From Catholic perspective, what is the meaning of the tearing of the veil at the death of Christ?

St. Thomas Aquinas himself thinks only the outer veil was rent; he says in his Commentary on St. Matthew's Gospel:

  And it ought to be observed that in the Temple there were two veils, namely, in the tabernacle, because there was the veil within the Holy of Holies, and there was another one, which was not in the Holy of Holies. And these two veils signify a twofold veiling, because the inside veil signifies the veiling of heavenly mysteries, which will be revealed to us: for then we shall be like to Him, when His glory shall have appeared. The other veil, which was outside, signifies the veiling of mysteries which pertain to the Church. Hence, the outer veil was rent, but the other one was not, to signify that mysteries which pertain to the Church were made known by Christ’s death; but the other veil was not rent, because heavenly secrets still remain veiled. Hence, the Apostle says: “But when Israel shall be converted to the Lord, the veil shall be taken away” (II Cor. 3, 16). Hence, by the Passion, all mysteries, which were written in the Law and the prophets, were opened, as it is stated: “Beginning at Moses and all the prophets, he expounded to them in all the scriptures the things that were concerning him” (Lk. 24, 27). Or the rending of the veil signifies the dispersion of the Jews. And because their glory was in the veil that was rent at the Lord’s Passion, it was signified that all glory was divided from them.

The Fathers seem to think only the outer veil was rent:

St. Thomas Aquinas, in his collection of the Fathers' commentaries on the Gospel, the Catena Aurea on Mt. 27, quotes these:

  St. Augustine: de Cons. Ev., iii, 19: The wording sufficiently shews that the veil was rent just when He gave up the ghost. If he had not added, "And, lo!" but had merely said, "And the veil of the temple was rent,: it would have been uncertain whether Matthew and Mark had not inserted it here out of its place as they recollected, and Luke had observed the right order, who having said, "And the sun was darkened," adds, "And the veil of the temple was rent in twain;" [Luke 23:45] or, on the contrary, Luke had returned to what they had inserted in its place.

  Origen: It is understood that there were two veils; one veiling the Holy of Holies, the other, the outer part of the tabernacle or temple. In the Passion then of our Lord and Saviour, it was the outer veil which was rent from the top to the bottom, that by the rending of the veil from the beginning to the end of the world, the mysteries might be published which had been hid with good reason until the Lord's coming. "But when that which is perfect is come," [1 Cor 13:10] then the second veil also shall be taken away, that we may see the things that are hidden within, to wit, the true Ark of the Testament, and behold the Cherubim and the rest in their real nature.

  St. Hilary: Or, The veil of the temple is rent, because from this time the nation was dispersed, and the honour of the veil is taken away with the guardianship of the protecting Angel.

The Catholic Haydock Commentary on Mt. 27:51 thinks both the inner and outer veils were rent (my emphasis):

  Ver. 51. The veil of the temple was rent. As there were in the temple two parts of the sanctuary, so there were two veils, or partition walls. The first sanctuary, called the holy, was separated by a veil from that part of the temple called the court of the Israelites. Into this outward sanctuary, called the holy, entered every day the priests that were in office. The second interior sanctuary, called the holy of holies, was also separated from the outward sanctuary by another veil. And into this holy of holies, no one was to enter except the high priest, and he but once a-year. Both these veils seem to have been rent at Christ's death: and by their being broken down, was signified first, that the ceremonies of the ancient law were to be abolished by the law of Christ; and also that heaven should be open to all. …


According to the Catholic Church, are Jews saved without faith in Jesus?

Paragraphs 35 and 36 of the document discussed by the news article you've linked to state (emphasis mine):

Since God has never revoked his covenant with his people Israel, there cannot be different paths or approaches to God’s salvation. The theory that there may be two different paths to salvation, the Jewish path without Christ and the path with the Christ, whom Christians believe is Jesus of Nazareth, would in fact endanger the foundations of Christian faith. Confessing the universal and therefore also exclusive mediation of salvation through Jesus Christ belongs to the core of Christian faith. So too does the confession of the one God, the God of Israel, who through his revelation in Jesus Christ has become totally manifest as the God of all peoples, insofar as in him the promise has been fulfilled that all peoples will pray to the God of Israel as the one God (cf. Is 56:1-8). The document "Notes on the correct way to present the Jews and Judaism in preaching and catechesis in the Roman Catholic Church" published by the Holy See’s Commission for Religious Relations with the Jews in 1985 therefore maintained that the Church and Judaism cannot be represented as "two parallel ways to salvation", but that the Church must "witness to Christ as the Redeemer for all" (No.I, 7). The Christian faith confesses that God wants to lead all people to salvation, that Jesus Christ is the universal mediator of salvation, and that there is no "other name under heaven given to the human race by which we are to be saved" (Acts 4:12).

From the Christian confession that there can be only one path to salvation, however, it does not in any way follow that the Jews are excluded from God’s salvation because they do not believe in Jesus Christ as the Messiah of Israel and the Son of God. Such a claim would find no support in the soteriological understanding of Saint Paul, who in the Letter to the Romans not only gives expression to his conviction that there can be no breach in the history of salvation, but that salvation comes from the Jews (cf. also Jn 4:22). God entrusted Israel with a unique mission, and He does not bring his mysterious plan of salvation for all peoples (cf. 1 Tim 2:4) to fulfillment without drawing into it his "first-born son" (Ex 4:22). From this it is self-evident that Paul in the Letter to the Romans definitively negates the question he himself has posed, whether God has repudiated his own people. Just as decisively he asserts: "For the gifts and the call of God are irrevocable" (Rom 11:29). That the Jews are participants in God’s salvation is theologically unquestionable, but how that can be possible without confessing Christ explicitly, is and remains an unfathomable divine mystery. It is therefore no accident that Paul’s soteriological reflections in Romans 9-11 on the irrevocable redemption of Israel against the background of the Christ-mystery culminate in a magnificent doxology: "Oh, the depth of the riches and wisdom and knowledge of God! How inscrutable are his judgments and how unsearchable his ways" (Rom 11:33). Bernard of Clairvaux (De cons. III/I,3) says that for the Jews "a determined point in time has been fixed which cannot be anticipated".

In summary, it appears that the document reiterates previously published Catholic documents that while the Jews are saved through Christ, their "participation" in the salvation of God is not a result of faith in Christ Jesus as Messiah but rather of the promises that God has made to them, which are irrevocable. Since this is contrary to the traditional understanding of salvation through faith and confession in Jesus Messiah, the document refers to the salvation of the Jews through Christ as a mystery. We continue to read in paragraph 37:

Here we confront the mystery of God’s work, which is not a matter of missionary efforts to convert Jews, but rather the expectation that the Lord will bring about the hour when we will all be united, "when all peoples will call on God with one voice and ‘serve him shoulder to shoulder’.

The document does reiterate the Vatican II position on organized mission work directed toward Jews in paragraph 40:

It is easy to understand that the so–called ‘mission to the Jews’ is a very delicate and sensitive matter for Jews because, in their eyes, it involves the very existence of the Jewish people. This question also proves to be awkward for Christians, because for them the universal salvific significance of Jesus Christ and consequently the universal mission of the Church are of fundamental importance. The Church is therefore obliged to view evangelisation to Jews, who believe in the one God, in a different manner from that to people of other religions and world views. In concrete terms this means that the Catholic Church neither conducts nor supports any specific institutional mission work directed towards Jews.

For a thorough exposition of the Roman Catholic doctrines concerning salvation, please read the article on that topic at CARM.
As for the history of relations between Catholicism and Judaism and the Roman Catholic Church's teachings, please read the Wikipedia article on that subject as a touchstone. It appears that the Catholic Church's teachings on the subject have been constant over the ages, and that Vatican II (1962 - 1965) was a turning point in the Vatican's position on the salvation of the Jews, as it was on many other doctrines. Specifically, the document titled Nostra Aetate deals with doctrines concerning the Jewish people. The Wikipedia article, which offers a brief historical background and a few modern considerations, states:

A new understanding of the relationship between Catholics and Jews is also reflected in the revised liturgy of Good Friday in a particular way. The pre-1962 version of the Good Friday Prayer of the Roman Rite had Catholics praying that the "perfidis Judaeis" might be converted to "the truth." The English cognate "perfidious" had, over the centuries, gradually acquired the sense of "treacherous." In order to eliminate misunderstanding on this point, Pope Pius XII ordered in 1955 that, in Catholic liturgical books, the Latin word "perfidis" be properly translated "unbelieving", ensuring that the prayer be understood in its original sense: praying for the Jews who remained "unbelieving" concerning the Messiah. Indeed, the same adjective was used in many of the ancient rituals for receiving non-Christian converts into the Catholic Church.

Here are excerpts from the two Good Friday prayers, before Vatican II,

Let us pray likewise for the unbelieving Jews: that the Lord our God may remove from their hearts the veil of unbelief: and that they may come to the knowledge of Jesus Christ our Lord.

...and after:

Let us pray
For the Jewish people,
The first to hear the word of God,
That they may continue to grow in the love of his name
And in faithfulness to his covenant.

The document under discussion is no new thing, but is meant to be a reflection upon Nostra Aetate on the fiftieth anniversary of its publication. Historically, the change in doctrine toward the salvation of the Jews represented by that document appears to be a recent and legitimate about-face from Tradition, since Church Fathers, Magisterial congregations, and Popes as recent as the twentieth century, have made clear statements that while the Jews should be especially protected from persecution, they can not be justified by their ancestry alone but only by conversion to the Universal Faith.
Again from Wikipedia,

Around 598, in reaction to anti-Jewish attacks by Christians in Palermo, Pope Gregory the Great (c 540–604) brought Augustine's teachings into Roman Law, by writing a Papal Bull which became the foundation of Catholic doctrine in relation to the Jews and specified that, although the Jews had not accepted salvation through Christ, and were therefore condemned by God until such time as they accept salvation, Christians were nevertheless duty-bound to protect the Jews as an important part of Christian civilization. The Bull said that Jews should be treated equitably and justly, that their property rights should be protected, and that they should keep their own festivals and religious practices. Thus, in the Papal States, Jews enjoyed a level of protection in law.


Have Creationists advanced any particular Theories of Origin that they claim are falsifiable via the scientific method?

Reasons to Believe is an organization which presents arguments in favor of Old Earth Creation, and on their Who We Are page, founder Hugh Ross states:

  One of the approaches I've developed, with the help of my RTB colleagues, is a biblical creation model that is testable, falsifiable, and predictive.

Their web site offers an overview of their "falsifiable" creation model approach, which includes:

  Using a model approach supplies researchers with enough detail to assist in further study. It offers explanations for how, when, where, and why the phenomena (or system) occurs. It anticipates, or “predicts,” discoveries that could either verify or falsify the model's explanation(s). The best models also yield specific suggestions for how near-future research may help improve understanding of the systems or phenomena they attempt to explain.

One example (the first I found) of a theory which they claim is falsifiable can be found in this article:

  These morphologically indistinguishable, yet genetically distinct ecomorphs can be properly considered as one of the many fingerprints that the Creator has left on His creation. In fact, if a single Creator was responsible for life, one could anticipate seeing repeated examples of the same blueprint throughout the biological realm. One would expect that a single Creator would reuse successful designs over and over again.

Note that this is not an academic article, so it's targeted at lay people, and as such the terminology is rather imprecise for my taste. It also seems to me that their claims of falsifiability are somewhat subjective, so I simply present them here as evidence that Creationists do, indeed, claim to produce falsifiable theories.


How do Christians reconcile Isaiah 42:4 with the death of Jesus?

This is easily reconciled because here is nothing to reconcile! Christ has never faltered, been disheartened, or crushed. He fulfilled His mission while here on earth, continues to do so as our advocate, and will continue to do so. His death and resurrection demonstrated this, rather than negating it!

A look at Christian commentaries shows that this is the common understanding among Christians.

  Matthew Henry's Concise Commentary

  Isaiah  42:1-4 This prophecy was fulfilled in Christ, Mt 12:17. Let
  our souls rely on him, and rejoice in him; then, for his sake, the
  Father will be well-pleased with us. The Holy Spirit not only came,
  but rested upon him, and without measure. He patiently bore the
  contradiction of sinners. His kingdom is spiritual; he was not to
  appear with earthly honours. He is tender of those oppressed with
  doubts and fears, as a bruised reed; those who are as smoking flax, as
  the wick of a lamp newly lighted, which is ready to go out again. He
  will not despise them, nor lay upon them more work or more suffering
  than they can bear. By a long course of miracles and his resurrection,
  he fully showed the truth of his holy religion. By the power of his
  gospel and grace he fixes principles in the minds of men, which tend
  to make them wise and just. The most distant nations wait for his law,
  wait for his gospel, and shall welcome it. If we would make our
  calling and election sure, and have the Father delight over us for
  good, we must behold, hear, believe in, and obey Christ.

Even in His darkest hour, as He prepared to fulfill His mission on the cross, He never faltered.  He did the will of the Father, putting aside His own needs without hesitation.

  Matthew 26:39  (KJV)

  And he went a little further, and fell on his face, and prayed,
  saying, O my Father, if it be possible, let this cup pass from me:
  nevertheless not as I will, but as thou wilt.

In my book. that's not faltering.  were it me, I'd have faltered. I'd have run away, lied, done anything to escape, yet He never faltered.  He went ahead, knowing what was to come, because He cannot falter.

Extra credit: Here's a list of sermons that preach on that verse, many of which focus on Christ as the triumphant servant.

Edits to address points missed per the comments below.  Namely, this comment:

  Maybe you know more than me and can show that there has been justice
  on this earth for the past 2000 years.

This was addressed in the very first paragraph on this answer, but I will expand upon it.  Repeating the relevant sentence, with some emphasis added:

He fulfilled His mission while here on earth, continues to do so as our advocate, and will continue to do so.

The misunderstanding here of commonly accepted Christian doctrine, is that in asking whether there has been justice on this earth in the last 2000 years is that His work was done on the cross.  That is not the case.  Christianity isn't based on His death, it's based on His resurrection, and the work that He continues to do.  He has still not, to this very second, faltered. He is still doing the will of his father.

The idea that the verse is nullified because perfect justice wasn't established on this earth by the time of His death ignores the very basic tenets of the Christian faith.  For there to be a discrepancy, His would would have had to end with His death.  It didn't.

Rather than going deep into detail on all of the doctrinal points that this question ignores, I'll simply summarize them with links for further study.

Christ was resurrected.  So His work didn't end at the cross.
He then spent a little more time on this earth after his resurrection, and then ascended to heaven.

Lest you wonder about the three days between His death and resurrection and somehow try to interpret that as a "pause", we know little of what He was doing during those three days, but we do know that He was active, "preaching to the spirits in prison".  (Although the exact details on what that means aren't clear.)

Christ continues, right now, even as I type this, to work as our Advocate.  (still hasn't stumbled.)

So, up until the present time, at no point has Christ stumbled, wavered, or strayed from his path.  We have no evidence whatsoever that he was discouraged, even in His darkest hour, or since.  Thus, the fact that perfect justice hasn't been established yet is not a problem.

Scripture does promise that He will establish new Heaven and a new Earth.  This is the point at which perfect justice will be established on earth, but that is a promise for the future.

Only if he stumbles before that future promise comes to pass could there possibly be any potential discrepancy here.


Calvinism: alternatives to evanescent grace?

Total depravity is not itself a zealous hatred of God, but an emphasis on the entire corruption of the whole person. Every faculty of the sinner has been corrupted by the fall. Nevertheless, unregenerate sinners hate God at different degrees, while regenerate sinners retain their sinful natures until glorification wherein there is a conflict between the new man and the old man, which explains why the regenerate still sin.
When we meet unregenerate people who by ordinary standards are nice people, we can be sure that God's grace is restraining their evil natures, and even recognize with thanksgiving the good that the unregenerate does at times, receiving it as it were by the hand of the Lord.
A falsely converted person may well have received real grace, just not regenerating grace. They would be all the more likely to receive these restraining graces because of their entry into the church where God's law will be taught and they will have godly examples from others.
Once clarifying our definitions, it appears the objection is now gone.


How has apostolic succession been "unbroken" in the wake of depositions, antipopes, immorality, and simony?

Apostolic succession and Papal succession are not one and the same.

Is the nature of apostolic succession meant to be an unbroken succession of valid popes?

No. Rather the nature of apostolic succession is meant to be an unbroken succession of valid bishops.

Not just the pope, but all Catholic bishops have unbroken apostolic succession. In other words, for a person to become a bishop (even the bishop of Rome - aka pope), he has to be consecrated (Laying on of hands) by another a validly consecrated bishop. Who in turn should have been validly consecrated by another validly consecrated bishop...... If we keep backtracking like this, we will end up with one of the thirteen apostles (The Twelve + Paul). This is considered to be unbroken line of apostolic succession.

Popes might validly succeed each other if and only if they have proper apostolic succession (i.e they are validly consecrated bishops). This is why Stephen II is not considered a pope.

The quote you have mentioned talks about continuity in the see of peter. That is the continuity in papal succession. In other words, there is no time in history of papacy where the see of peter (or the post of the pope) is deserted with no claimants and no conclave was held to elect his successor. Note that the see could be vacant, but during that time the conclave could be taking place. This is what is called unbroken line of successors of Peter.

Instances of popes being involuntarily deposed

New popes are not considered valid unless the previous pope submits to his authority. (As in deposing of Pope John XII and Pope Leo VIII's invalid first election)

Instances of contention over the papacy by two or even three popes simultaneously

Even though at the time when this was happening, it was unclear as to who is the legitimate successor, we can be sure that at least one of them was legitimate. In other words, the see of peter was not vacant. (Retrospectively we can determine the legitimate successor.)

Instances of nepotism and dynastic possession ... popes engaging in orgies, conspiracies, accepting bribes ....

Sinfulness of the person holding the office does not invalidate the office itself.

Instances of the papal office being sold.

If a reigning pope resigns for money, it is a sin (Simony). Like all sins this is also a voluntary act. Hence he will cease to be pope.


Carrying the cross: a profane act or religious act?

The procession of the Way of the Cross is an important Catholic tradition, so any criticism of a transgender person taking part in such a procession can only be based on the person's gender identity. However, it is possible for Christians to object to participation in an actual mock crucifixion, and the evidence for this is mixed.

Crucifixion in the Philippines is a devotional practice held every Good Friday, where the custom is said to be strongly discouraged by the Catholic Church in the Philippines. How 'strongly' is unclear, since the Church has not stamped out the practice even in urban centres.

A school in Brazil has had very young children enact crucifixions for Holy Week complete with fake wounds and a crown of thorns. Although the children suffered no physical harm, the practice has been condemned by some as akin to manipulation, but the story has also attracted many Facebook 'likes'. What is interesting here is that this kind of play-acting by children can only encourage adults to act out crucifixion, yet there is no evidence the Church hierarchy has condemned it.

The bishops of Brazil have, however, condemned a Brazilian sports magazine for printing photo-shopped photos of famous soccer star Neymar, in which he appears to be crucified. Their concern seems to be more with the inappropriate use of religious images than with the inappropriateness of mock crucifixion.

A symbolic reenactment of the crucifixion of Christ performed in Brazil at World Youth Day, attended by Pope Francis, can only encourage those who believe these reenactments have papal support.

The news item suggests that even the gay parade itself drew condemnation. There is no suggestion that those taking part were not as sincere as other Christians taking part in Way of the Cross parades on Good Friday, so the objection appears to be to gays openly parading their Christian faith, rather than a suggestion they were mocking the faith. The actual reenactment by a transgender person drew a strong response, including a death threat. If there were evidence that others who have taken part in reenactments in Brazil have been denounced in the same way, then we could say the objections were grounded on objections to profane performances, but this is not the case.

The position of the Catholic Church is ambiguous. On the one hand, Filipino bishops frequently object to reenactments in the Philippines, on the other hand, we are told that a crucifixion reeneactment was part of the entertainment at a World Youth Day attended by Pope Francis, in Brazil.

The Baptist and Presbyterian Churches are not so well represented in Brazil, but here we see a fairly casual non-denominational parade that takes place in Australia every year. Churches participating in the Walk for the Cross were St. Giles Presbyterian Church, The Seventh Day Adventist, St. John The Baptist Anglican Church, St. Joseph's Roman Catholic Church, and St. Matthew Wesley United Church. There is no suggestion that the Walk ends in a mock crucifixion.


Who is an "enemy" in the context of this video?

The speaker is Michael Oh, the founder of Christ Bible Institute Japan. He is of Korean heritage but grew up in America.

I found a video (plus transcript) that explains more fully his animosity toward Japan, and the process of forgiveness he underwent. The video is of a talk he gave on night 5 of the Urbana missions conference in 2009. The animosity basically comes from what he calls the "Asian holocaust" and particularly Japan's occupation of Korea, in which his family experienced much cruelty at the hands of the Japanese.

In the video, Oh mentions these general facts about Japanese war crimes:

"According to historian Chalmers Johnson as many as 30 million Filipinos, Vietnamese, Cambodians, Indonesians, Koreans, Chinese, and others were slaughtered in Asia during the years of Japanese aggression leading up to and including World War II."
"Japanese scientists tested various chemical and biological weapons such as bubonic plague and anthrax on human victims."
"Women were impregnated by soldiers and doctors, their bellies sliced open, their babies removed, and then experimented upon," which seems to refer to findings of the Women's International War Crimes Tribunal on Japan's Military Sexual Slavery.
"So cruel were Japanese medical experiments that Nazi scientists who visited the facilities vomited from the horror of what they saw."
"Almost 200,000 Korean women, and girls as young as 12 years old, were subject to rape upwards of 100 times per day."

He also mentions that his father was beaten if he used his Korean name and that his grandfather's sister was married off after his family discovered that the Japanese took unmarried girls as comfort women.

I think you're overthinking the general application of "love your enemies," which was clearly intended for all enemies of all kinds. That is, the Christian is supposed to love all who mistreat him; this is the call that Michael Oh responded to by becoming a missionary to Japan. The command is recorded in Matthew 5 and Luke 6. Luke's version (in the NET) says:

  Love your enemies, do good to those who hate you, bless those who curse you, pray for those who mistreat you. ... If you love those who love you, what credit is that to you? For even sinners love those who love them. And if you do good to those who do good to you, what credit is that to you? Even sinners do the same. ... But love your enemies, and do good, and lend, expecting nothing back. Then your reward will be great, and you will be sons of the Most High, because he is kind to ungrateful and evil people. Be merciful, just as your Father is merciful.


Does transubstantiation apply to the Last Supper, which occurred prior to the crucifixion?

Yes, this is the teaching of the Catholic Church.
The Catechism of the Catholic Church states:

1366 The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit:
[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.

The Eucharist according to Catholicism is not just a memorial or symbol, as many Protestants believe, but that the sacrifice of Christ on the cross is truly and actually made present; it is not merely in the past.
In fact, the doctrines concerning the Eucharist seem to be known because of the offering at the Last Supper; if it had not been a real sacrifice, the teachings concerning transubstantiation would likely never have existed:

On that occasion He likewise made an unbloody food-offering, only that, as Antitype, He accomplished something more than a mere oblation of bread and wine, namely the sacrifice of His Body and Blood under the mere forms of bread and wine. Otherwise, the shadows cast before by the "good things to come" would have been more perfect than the things themselves, and the antitype at any rate no richer in reality than the type. Since the Mass is nothing else than a continual repetition, commanded by Christ Himself, of the Sacrifice accomplished at the Last Supper...
Matthew 26:28 For this is My blood of the new covenant, which is shed for many for the remission of sins.
(also see Mark 14:24, Luke 22:20)

And again, Catholic Encyclopedia says concerning the Real Presence of Christ:

The Church's Magna Charta, however, are the words of Institution, "This is my body — this is my blood", whose literal meaning she has uninterruptedly adhered to from the earliest times. The Real Presence is evinced, positively, by showing the necessity of the literal sense of these words, and negatively, by refuting the figurative interpretations.
... [they did not claim the bread and blood to be signs of Christ's body, but] in the original text corpus (body) and sanguis (blood) are followed by significant appositional additions, the Body being designated as "given for you" and the Blood as "shed for you [many]"; hence the Body given to the Apostles was the self same Body that was crucified on Good Friday, and the Chalice drunk by them, the self same Blood that was shed on the Cross for our sins, Therefore the above-mentioned appositional phrases directly exclude every possibility of a figurative interpretation.

The Catholic teaching is that Christ did not claim the body and bread to be a metaphor for His body and blood; He did not say, "this bread is my body," but said "this is my body." He was crucified once for all time -- past, present, and future. Just as He provides salvation to people who lived prior to His sacrifice, so He can be present in the institution of the Eucharist prior to His crucifixion.

... so, when He said: "Take ye, and eat. This is my body", the Apostles received from the hand of the Lord His Sacred Body, which was already objectively present and did not first become so in the act of partaking. This non-dependence of the Real Presence upon the actual reception is manifested very clearly in the case of the Chalice, when Christ said: "Drink ye all of this. For [enim] this is my Blood." Here the act of drinking is evidently neither the cause nor the conditio sine qua non [necessary precondition] for the presence of Christ's Blood.


Do any Christian denominations prohibit eating rare meat?

If you google "Torah Christian", you may not find a denomination, per se, but you'll see several sites which support the notion that Christians must follow the Torah.

I have a friend who belongs to such a church (they're all "non-denominational", but loosely associate with each other) who maintains that Christians should not eat rare steak because there is still blood in it.

The problem with this stance is that, unless you are eating truly raw meat (eg off a freshly-killed rabbit before any cooking), there is [practically] no blood left in the meat, it's a combination of water and myoglobin.

Myoglobin is functionally similar to hemoglobin, but stores oxygen in muscles vs blood, respectively.

There are all kinds of reasons to eat (or not eat) rare steak - that "blood" is still in it is not one of them.


Is there biblical support for "feeling the presence of God" as reported by multiple Christian conversion testimonies?

The late itinerant preacher, evangelist, and author, Paul E. Little, who traveled nationally as a representative of Inter/Varsity Christian Fellowship, said in a sermon I listened to at spiritual emphasis week at Emmaus Bible College, that Christians should not necessarily experience a "liver quiver" when attempting to discern God's will. Good advice.
Feelings come and go. Jesus's disciples likely experienced many liver quivers as they observed their Lord and Savior in action, particularly during his more spectacular signs and wonders (e.g., calming the stormy Sea of Galilee with a simple command, being with him at the Transfiguration where Moses and Elijah joined them, and the raising of Lazarus from the dead).
Both then and today, however, believers are to walk by faith and not by sight--or by feelings, for that matter (see 2 Corinthians 5:7)!
I have a friend, a believer in Jesus, who heard the voice of God at his conversion. Talk about "liver quivers"! The feelings that that voice elicited from my friend were unforgettable for him, but they added not a whit to the reality of his conversion.
Imagine the apostle Paul at his conversion (when he was still Saul). Imagine the feelings that Jesus's voice elicited from him, not to mention his reaction to the holy light which left him blind, temporarily. As spectacular as his conversion was, I imagine the newness, novelty, and emotionality of what occurred to Saul on the road to Damascus eventually wore off. As Paul grew into his faith, the conviction that his calling by God was real, deepened. What became increasingly important to him was being obedient to his calling by God to be an evangelist to the Gentiles.
In short, feelings are great. They are, after all, God-given. They are part of God's design of us as creatures made in his image. Feelings of ecstasy, like a drug-induced  high (I imagine!), are bound to fade in time. If they are not superseded by belief, faith, and trust, however, they are of little value in the long run. If God graciously gives us that warmth of healing so many Christians describe, then great. If he graciously withholds such feelings from us, sometimes that can ultimately be even better for our walk of faith.


While it's not always the most reliable source for knowledge, a quick trip to Wikipedia's article on Transcendental Meditation gets us a bit of well-referenced history on the whole Transcendental Meditation movement.  Included in the article is this:

  In the 1950s, the Transcendental Meditation movement was presented as
  a religious organization. The Transcendental Meditation technique was
  held to be a religion in a New Jersey court case. By the
  1970s, the organization had shifted to a more scientific presentation
  while maintaining many religious elements. The movement now
  describes itself on a spiritual, scientific, and non-religious basis.
  This shift has been described by both those within and outside the
  movement as an attempt to appeal to the more secular West.

I included the above because of your statement: "It is said that it isn't a religion and people of all religions participate in this practice.", and the fact that at least part your question seems to be centering around the question of "Would participating in this be participating in another religion".

The movement absolutely had it's roots in Hinduism, so on the one hand, there is cause for scrutiny.

However, simply meditating... Is that the same as worshiping Hindu gods? I don't know that I'd be dogmatic on that.  This appears to be one of those gray areas, and ultimately, this is something that you will have to decide, and it will be between you and God.

I come from a denomination that has members that believe that everything even remotely related to evil has no place in a Christian's life.  Harry Potter?  Satan's attempt to introduce our kids to witchcraft as harmless...  Rock music? Satan's tool for stirring up anger.  Dance? Satan's tool for stirring up lust.

To them, the idea of participating in Transcendental Meditation is no different than worshiping idols.  Me, I'm not so sure.  I personally don't see that practicing a meditation technique is the same as converting to Hinduism or worshiping their gods.  The technique ma be based on Hindu teachings, but that does not mean that the technique itself is Hinduism.

This may be far-fetched, but as a comparison - suppose someone were to come up with a cure for cancer and the person who came up with the treatment did so based on teachings of a Wiccan herbalist.  Would taking that cure be the same as participating in Wicca?  I don't think so.

Your question also includes the following: "Does transcendental meditation conflict with any Christian beliefs?"

That is going to be next to impossible to answer, because there are a wide variety of Christian beliefs.  Again, there are plenty in my denomination that would say "yes".  And there would be plenty that would say "no."

The real questions here are:

Does anything in TM conflict with your Christian beliefs.

and

Should I participate in it?

We can't answer the first one for you at all.  You should know what you believe.  If you're unsure of what you believe or should believe, then I'd focus on that first.  Once you know what you believe, you could study about TM yourself and keep your eye out for patently non-Christian elements.  Knowing what non-Christian or even anti-Christians teach can be beneficial.  At any rate, you'll need to make your own mind up on that.

As for the second question, there is precedent in Scripture, which may completely nullify my answer to the first question.

  Romans 14:14-23 (KJV) 14 I know, and am persuaded by the Lord Jesus, that there is nothing unclean of itself: but to him that
  esteemeth any thing to be unclean, to him it is unclean.

  15 But if thy brother be grieved with thy meat, now walkest thou not
  charitably. Destroy not him with thy meat, for whom Christ died.

  16 Let not then your good be evil spoken of:

  17 For the kingdom of God is not meat and drink; but righteousness,
  and peace, and joy in the Holy Ghost.

  18 For he that in these things serveth Christ is acceptable to God,
  and approved of men.

  19 Let us therefore follow after the things which make for peace, and
  things wherewith one may edify another.

  20 For meat destroy not the work of God. All things indeed are pure;
  but it is evil for that man who eateth with offence.

  21 It is good neither to eat flesh, nor to drink wine, nor any thing
  whereby thy brother stumbleth, or is offended, or is made weak.

  22 Hast thou faith? have it to thyself before God. Happy is he that
  condemneth not himself in that thing which he alloweth.

  23 And he that doubteth is damned if he eat, because he eateth not of
  faith: for whatsoever is not of faith is sin.

and

  1 Corinthians 8:4-13 (KJV)
  4 As concerning therefore the eating of those things that are offered in sacrifice unto idols, we know that an
  idol is nothing in the world, and that there is none other God but
  one.

  5 For though there be that are called gods, whether in heaven or in
  earth, (as there be gods many, and lords many,)

  6 But to us there is but one God, the Father, of whom are all things,
  and we in him; and one Lord Jesus Christ, by whom are all things, and
  we by him.

  7 Howbeit there is not in every man that knowledge: for some with
  conscience of the idol unto this hour eat it as a thing offered unto
  an idol; and their conscience being weak is defiled.

  8 But meat commendeth us not to God: for neither, if we eat, are we
  the better; neither, if we eat not, are w
  e the worse.

  9 But take heed lest by any means this liberty of yours become a
  stumblingblock to them that are weak.

  10 For if any man see thee which hast knowledge sit at meat in the
  idol's temple, shall not the conscience of him which is weak be
  emboldened to eat those things which are offered to idols;

  11 And through thy knowledge shall the weak brother perish, for whom
  Christ died?

  12 But when ye sin so against the brethren, and wound their weak
  conscience, ye sin against Christ.

  13 Wherefore, if meat make my brother to offend, I will eat no flesh
  while the world standeth, lest I make my brother to offend.

Participating in TM is very comparable to eating food sacrificed to idols in this regard.
Paul is teaching that eating food sacrificed to idols in and of itself has no effect on the saved Christian.  The Christian isn't worshiping the idol, and meat is meat. (1 Cor 8:4-8)  Similarly simply meditating using TM isn't the same as practicing Hinduism.  Meditation and relaxation techniques are just meditation/relaxation techniques.

However, reading on in 1 Cor 4: 9-13, you need to be aware that people are watching you.  There may be non-Christians who are watching your actions.  Maybe people you've witnessed to, and if they see you participating in TM, they may think that there's nothing at all wrong with it, either.  They, unlike you , may not know what is Christian and what is not.

Another good comparison would be martial arts.  I was in martial arts of varying types when I was younger.  Almost all of the martial artists I hung out with fancied themselves to be Zen Buddhists to one degree or another.  If they weren't quoting Bruce Lee, they were quoting Zen Buddhist sayings.  Some of them went further and got sucked in completely, some were just posers.  As a Christian, I feel confident that I could practice martial arts again, without getting sucked into the eastern mysticism that tends to go hand-in-hand with the martial arts culture.

TM is pretty similar.  I feel pretty confident that I could sort out the mysticism, determine what is patently un-Christian (if anything), and participate with a clean conscience.

However, I know plenty of people that wouldn't be able to sort it out.  non-Christians, or those that are Christians in name only (raised in a Christian home, and somehow they think this is enough).  They might say "Hey, if he can do it, so can I.  They could go in and get sucked in, because they don't have the background and knowledge that I have.  (That isn't meant to sound as arrogant as it does.  I just mean I know what I believe, and some haven't given it much thought.)

So, personally, for their sake I wouldn't participate.  This is exactly what Paul was talking about in 1 Corinthians.


What does 'first estate' mean in Jude 1:6 (KJV)?

The Strong's for this is:

  archē
  ar-khay'
  From G756; (properly abstract) a commencement, or (concrete) chief (in various applications of order, time, place or rank): - beginning, corner, (at the, the) first (estate), magistrate, power, principality, principle, rule.

Sounds like it means they gave up their rank or position, in addition to their "habitation" (residence) in exchange for some wicked thing they desired. Now, as a result, they are enjoying their 2nd estate--in chains.


What is the Reformed Protestant understanding of God's Permissive Will, His Sovereign or Decretive Will and Efficacious Will?

Your quote source is part of R.C. Sproul's explanation of Permissive Will, so you will find the context by reading the complete article titled "Discerning God’s Will: The Three Wills of God".  After a quick and relatively straightforward definitions of Decretive and Prescriptive wills, he then warns us of "the faulty distinction between willing and permitting" by quoting Calvin along with the Biblical support Calvin used for his thesis.  Sproul then ends with a quote from St. Augustine.

A related resource I found is a free book by R.C. Sproul "Can I Know God's Will" which I could also find the PDF here where he also discussed Decretive and Preceptive Will of God but introduced "God's Will of Disposition" as roughly equivalent to his "permissive" will, similar to how this article also divide God's will into the same 3 categories.  That's in chapter 1 ("The Meaning of God's Will") to which he adds chapter 2 ("The Meaning of Man's Will") followed by 2 application chapters on job and marriage.

Other related resources I think you'll find valuable are:

Nathaniel's answer to a related question which gives you the relevant Biblical verses for the Decretive and Preceptive Will of God based on R.C. Sproul and Charles Hodge
Charles Hodge's treatment of The Will of God in his famous (although dated) 1871 Systematic Theology Volume 1 Chapter V Section 9
Blog article "A Theology Of the Will of God by Adam Setser, a Southern Baptist theology student, surveying different ways (other than decretive vs. preceptive) to view the two sides of the will of God within the Reformed tradition, nicely paraphrasing some of Hodge's treatment in 2015 English.

I found Adam Setser's treatment to be the most helpful to address your confusion because he goes into the Latin words of 5 different pairings of tensions that Reformed theologians have identified over the centuries.  I put in bold the terms you are including in your question.  From the analysis below, I think the confusion comes from mixing terms from different pairings.

The backbone: voluntas beneplaciti (will of desire) / voluntas decreti (will of decree) / voluntas occulta (secret will)  VS.  voluntas signi (signifying will) / voluntas revelata (revealed will) / voluntas praecepti (will of precept) (see subsection C in Hodge)
Choice: voluntas absoluta VS. voluntas conditionalis (see subsection E in Hodge)
Causation: voluntas antecedens VS. voluntas consequens (see subsection D in Hodge)
Theodicy: voluntas efficiens (efficient will) VS. voluntas permittens (permitted will)

  If God’s will must account for all creation, what about evil, Satan, and hell?

  Voluntas efficiens: the “efficient will” refers to those aspects of His will who receive His full affirmation. Efficiens also means to create, cause, or produce. So this is the product of His creation which obeys Him perfectly and gets his affirmation.

  Voluntas permittens: the “permitted will” refers to the part of creation which doesn’t get His affirmation. He still has willed its existence, but He isn’t pleased with it. He permits it to exist because it serves His purposes (e.g., Satan).

Efficacy: voluntas efficax (effective will, efficacious) VS. *voluntas inefficax** (ineffective, HERETICAL!)

If the above terms are not enough, there are a few more distinctions in the famous Reformed Systematic Theology textbook (1938) by Louis Berkhof: Part One (Doctrine of God), THE BEING OF GOD Chapter VII (The Communicable Attributes), Section D (Attributes of Sovereignty), pages 82-87, HTML version here, PDF here.  These 6 pages I think will be the best resource to integrate all the above distinctions into a coherent whole within the Reformed tradition where Dr. Berkhof said how the antecedent/consequent and absolute/condition pairings found little favor in Reformed theology while the decretive/preceptive, eudokia/eurestia, beneplacitum/signum, and secret/revealed pairings were more generally accepted. He discusses the issue of "permissive will" in a subsection titled "God's will in relation to sin".  Finally by necessity of its being a textbook Dr. Berkhof's treatment also adds contrast between Reformed understanding of the sovereign power of God to some modern non-Reformed positions (Strauss, Schleiermacher).

While the above resources are "certifiably" Reformed in the theology (despite different theologians, e.g. Dr. Berkhof and Dr. Hodge, organizing the discussion of God's will differently), I think the following article titled "Providence of God" from the 1997 Baker's Evangelical Dictionary of Theology should provide you with a larger context to place the will of God in a broader as well as more ancient perspective, contrasting how in OT, 2nd Temple, and NT traditions Jewish and Christian believers view God's will in PERSONAL terms rather than IMPERSONAL fate / deterministic way that the Greco-Roman philosophy adherents (such as Stoics) view God's will.

One way to consider God's will in an even larger context is of course by studying Louis Berkhof's textbook treatment of THE WORKS OF GOD where he begins by discussing The Divine Decrees in General (Chapter I), Predestination (Chapter II), Creation (Chapters III to V), and ends with Providence (Chapter VI).


What is meant by "men have been raised up from the fall to far greater heights?"

From the Catechism of the Catholic Church:

  CCC 412 But why did God not prevent the first man from sinning? St. Leo the Great responds, "Christ's inexpressible grace gave us blessings better than those the demon's envy had taken away." and St. Thomas Aquinas wrote, "There is nothing to prevent human nature's being raised up to something greater, even after sin; God permits evil in order to draw forth some greater good. Thus St. Paul says, 'Where sin increased, grace abounded all the more'; and the Exsultet sings, 'O happy fault,. . . which gained for us so great a Redeemer!'".

In brief, in Eden we had no Saviour, the Grace had not been shown to the fullest, and our "greater good" was out of reach.


Are there any Bible that comes with explanations of each verses?

Explanations of the Bible are called commentaries, and there are lots of them available. Some commentaries are focused on a specific book or section of the Bible, and some comprehensive commentaries attempt to cover the whole Bible.

Of course, these commentaries are written by people, and as such the explanations are from the theological viewpoint of whoever wrote it. You can select commentaries based on your preferred denomination, or you can read multiple commentaries and learn multiple views on the scripture.

Study Bibles are Bibles with commentary built-in. Usually these include notes at the bottom of each page with commentary on whatever verses are on that page. A study Bible can be very useful for getting a quick explanation of a verse that is giving you trouble. But again, this commentary is written by a person, so it is not infallible and contains a denominational bias, so keep that in mind as you read, and consult other commentaries as part of your study.


What does it mean to "worship in truth"?

So to worship in spirit means to do so with full purpose of heart.

  (KJV) Matt 15:8-9 This people draweth nigh unto me with their mouth, and honoureth me with their lips; but their heart is far from me. 9) But in vain they do worship me, teaching for doctrines the commandments of men.

The Lord was speaking about the Pharisees here who outwardly followed all of the commandments of the law of Moses, but inwardly they were vile sinners. Later in the chapter Christ talks about how it is not what goes into us that defiles us (talking about the law of Moses restrictions on food and washing before eating) but what comes out of our hearts. So to worship in truth is to not only worship outwardly but to also worship in your spirit and heart because that is what truly defines a man's belief.


Why couldn't a person in the OT be both priest and king?

The strongest restriction is that the priests of the Israelite nation could only be from the tribe of Levi, and specifically on descendants of Aaron. Some verses showing this include Numbers 3:10, 3:38 and 16:40:

  ... This was to remind the Israelites that no one except a descendant of Aaron should come to burn incense before the Lord ...

The kingship was not so strongly restricted, but Genesis 49:10 does clearly prophesy that it belongs to Judah:

  The scepter will not depart from Judah, nor the ruler’s staff from between his feet ...

So looking at the various people you asked about:

Melchizedek: was not an Israelite, so neither of these restrictions apply to him. He was a faithful foreign king.

Jesus: was an Israelite and is King because he is descended from David. He isn't a descendant of Aaron and his priesthood isn't from them. Instead he is a priest because he became a man and died on the cross for humanity.

Saul: was not a Judahite or a Levite. He definitely had no right to act like a priest, but the Genesis 49 prophesy was not so strict, so God wasn't being contradictory when he appointed Saul as king.

Kings were mentioned in the Law, most explicitly in Deuteronomy 17:14-20. That passage describes how the kings were to act, and it explains even the best kings like David and Solomon got into trouble. It doesn't say however which tribes the kings could come from.


What does it mean for Christ to have been slain before the creation of the world?

Notice the reading of the King James Version:

  Revelation 13:8 (KJV) And all that dwell upon the earth shall worship
  him, whose names are not written in the book of life of the Lamb slain
  from the foundation of the world.

Modern Translations read differently:

  Revelation 13:8 (ESV) and all who dwell on earth will worship it,
  everyone whose name has not been written before the foundation of the world in the book of life of the Lamb who was slain.

  Revelation 13:8 (NLT) And all the people who belong to this world
  worshiped the beast. They are the ones whose names were not written
  in the Book of Life before the world was made--the Book that
  belongs to the Lamb who was slaughtered.

  Revelation 13:8 (NASB) All who dwell on the earth will worship him,
  everyone whose name has not been written from the foundation of the world in the book of life of the Lamb who has been slain.

It seems that modern translations show that the Lamb was not slain before the foundation of the world.


The key issues seem to be in the Code of Canon Law, Chapter VI, canon 1125, which says in part:

1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;
2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;
3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.

A theme common to all these clauses is an attempt to maintain the Catholic partner and all future children as members of the Catholic faith, whether out of concern for their souls or concern for Church membership. To ensure that the Catholic faith is seen by both partners as paramount and that they regard their marriage as an exclusively Catholic sacrament, canon 1127, clause 3, says:

It is forbidden to have another religious celebration of the same marriage to give or renew matrimonial consent before or after the canonical celebration according to the norm of §1. Likewise, there is not to be a religious celebration in which the Catholic who is assisting and a non-Catholic minister together, using their own rites, ask for the consent of the parties.

Until around the middle of the twentieth century, the Catholic Church tended to insist the non-Catholic partner convert to Catholicism and only permitted mixed marriages as a last resort, in cases where conversion was refused and the Catholic partner was at risk of marrying outside the Church. Which brings us to the second part of the question - the fact that Catholics were increasingly prepared to enter into mixed marriages, with or without Church approval, brought about a relaxation of this impediment. Now, there are many Catholics who don't think this is an issue and are willing to make the required undertakings merely as a formality, or to marry outside the Church. Marrying outside the Church might mean the Catholic is no longer able to receive communion, but worldwide statistics from 2004 show that in many nominally Catholic countries, a minority of the population even goes to mass every week.
Apart from dispensation to marry outside the faith, subject to the above requirements, it is not possible to provide evidence against this 'belief', because those requirements are not a belief but Church doctrine. Something that is Church doctrine is its own proof.


If a Mormon man turns from Mormonism, what happens to his Mormon wife when she dies?

Perhaps some things should be clarified a bit here first.
First part of your question:

I'm told that according to Mormon doctrine, a Mormon woman has to be granted entry into heaven by her Mormon husband. Assuming my impression is correct, ...

Not really.
Unmarried women can certainly get to heaven -- and depending on what you mean by "heaven," they don't even have to be Mormon. Remember, Mormons believe what Paul teaches about the three degrees of glory in 1 Corinthians more literally than most Christians, in that there are three main kingdoms, all of which are called "heaven."
Unmarried men and women can even obtain the highest kingdom. However, in the Doctrine and Covenants, section 131, more is revealed about this state (the celestial glory):

1 In the celestial glory there are three heavens or degrees;
2 And in order to obtain the highest, a man must enter into this order of the priesthood [meaning the new and everlasting covenant of marriage];
3 And if he does not, he cannot obtain it.
4 He may enter into the other, but that is the end of his kingdom; he cannot have an increase.

To obtain this highest glory is called exaltation. It is living like God does, being as He is.
Eternal marriage (also called a sealing), faithfully kept by both individuals, happens to be that final requirement.
I suppose then, that what you mean by "going to heaven" is actually exaltation.
So, on to the next part of your question:

... what would happen if the husband turned from Mormonism and then died?

He would face the consequences in the hereafter on his own. It's not even so much about turning from "Mormonism" at that point, but rather about breaking your covenants made in the temple, to God and to your wife, for which he would need to repent.

Would the wife need to quickly remarry a Mormon before she died in order to get to heaven?

No. Again, assuming you mean, "in order to obtain exaltation": LDS members are taught and reminded (a few sources below) that all who are faithful here will have everything and more made up after this life, and all who do not have the opportunity to be eternally married here will be able to be sealed to a faithful husband there, if that be their desire.
The eternal nature of covenants performed in the LDS temples means that whether they are made in this life or the next, they are in effect for eternity (cf. parable of the laborers in the vineyard).
Sources and further reading:

Gospel Principles, Chapter 38: Eternal Marriage "all Heavenly Father’s children who are faithful to their covenants in this life will have the opportunity to receive all the blessings of the gospel in the eternities, including the opportunity to have an eternal family."

The Eternal Blessings of Marriage (a recent General Conference address) "All things are possible to the Lord, and He keeps the promises He inspires His prophets to declare. Eternity is a long time. Have faith in those promises and live to be worthy of them so that in His time the Lord can make them come true in your life. With certainty, you will receive every promised blessing for which you are worthy."

To the Single Adult Sisters of the Church "I also recognize that not all women in the Church will have an opportunity for marriage and motherhood in mortality. But if those of you in this situation are worthy and endure faithfully, you can be assured of all blessings from a kind and loving Heavenly Father—and I emphasize all blessings. I assure you that if you have to wait even until the next life to be blessed with a choice companion, God will surely compensate you. Time is numbered only to man. God has your eternal perspective in mind."


Trinity (Christians) = Trimurti (Hindus)?

One could do a great deal worse than quote another answer:

  There are several heresies that one needs to be careful of when discussing a topic such as this. For example, Modalism which declares that God is not three distinct persons, but that He merely reveals himself in three different forms. Or, Arianism which declares that Christ and the Holy Spirit are creations of the Father and not one in nature with Him. Or Partialism which declares that the Father, Son, and Holy Spirit are not distinct persons of the Godhead, but are different parts of God each composing a third of the divine.

Each of the different views of the Trimuti within Hinduism falls into one or more of those camps. My analysis (based on Wikipedia) may not be complete, but I posit this:—

Sauram: modalism
Vishaivism: Arianism
Shavism and Smartism: partialism

So: no, apart from both the Trinity and Trimurti being "three", there are no similarities. Specifically, the Trimurti denies that the three gods are wholly three and wholly one.


What is the Old Testament basis for the position that God is a God of love, compassion, and mercy?

There are a host of verses to which one could look throughout all of the Old Testament, with the first being in Genesis 3:8-21, God could've brought complete destruction on Adam and Eve, but instead showed mercy by allowing them to live and gave them a promise of a redeemer who would eventually come to destroy the serpent.

Genesis 4:6-7 shows God having compassion on Cain before he killed his brother:

  Then the LORD said to Cain, "Why are you angry? And why has your countenance fallen? If you do well, will not your countenance be lifted up? And if you do not do well, sin is crouching at the door; and its desire is for you, but you must master it."

Jump ahead to the conquest of Canaan, and see the mercy and love shown to Rahab and her entire family (she is so important for having been saved in Jericho that she is in the human line of Christ):

  "Now therefore, please swear to me by the LORD, since I have dealt kindly with you, that you also will deal kindly with my father's household, and give me a pledge of truth, and spare my father and my mother and my brothers and my sisters, with all who belong to them, and deliver our lives from death.? So the men said to her, "Our life for yours if you do not tell this business of ours; and it shall come about when the LORD gives us the land that we will deal kindly and faithfully with you."

She's even celebrated in the New Testament, "By faith Rahab the harlot did not perish along with those who were disobedient, after she had welcomed the spies in peace."

In Daniel 3:28-30, we see God's deliverance of Shadrach, Meshach, and Abednego as having a profound impact on the king:

  Nebuchadnezzar responded and said, "Blessed be the God of Shadrach, Meshach and Abed-nego, who has sent His angel and delivered His servants who put their trust in Him, violating the king’s command, and yielded up their bodies so as not to serve or worship any god except their own God. Therefore I make a decree that any people, nation or tongue that speaks anything offensive against the God of Shadrach, Meshach and Abed-nego shall be torn limb from limb and their houses reduced to a rubbish heap, inasmuch as there is no other god who is able to deliver in this way." Then the king caused Shadrach, Meshach and Abed-nego to prosper in the province of Babylon.

In Esther we see God strategically putting one of His people into a position of extreme influence to the King whereby He saves His people from the maniacal plans of Haman who wanted to exterminate the Jews.

In II Kings 5, we see God having mercy on the captain of the Assyrian army, Naaman via the instrument of a Hebrew maid girl in his house.

In short, the Old Testament is as replete with God's mercy, love, and compassion as the New Testament.


Yes, there are lots of examples! There are still professional Iconographers who go around painting Iconography for a living!
In the Orthodox Church the Temples are traditionally fully covered in frescoes.
Here is an example:

Since this is a great part of the Orthodox Church, there are many people who have been trained in this Holy Art and are going around the world adorning Temples.

In fact, at this very moment, the church which I am going to is having it's hitherto white and barren walls covered in frescoes! An iconographer from Serbia is doing all the painting of the frescoes throughout the whole church. At the moment, though, only slightly more than the dome is done.

Here is a snapshot of the dome from a little while ago:

There is also a slightly outdated video of the Iconographer at work which can be seen here.

And if you are really interested, every now and then, some very good Iconographer will hold an Iconography work-shop... I have never attended one, but from what I hear they are very fun.

I even found a link on OrthodoxWiki which has a list of Modern Iconographers.

Also, probably of great interest to you, is the website of one Iconographer, Vladimir Grygorenko. Take a look at his portfolio if you want to know the answer to this question...

And, a little more which probably won't be useful to you, here is a link which gives the Orthodox Iconographer some rules and prayers as guidance.
One of the prayers is as follows:

  A Prayer Before Beginning an Icon:

  O Divine Lord of all that exists, Thou hast illumined the Apostle and Evangelist Luke with Thy Holy Spirit, thereby enabling him to represent Thy most Holy Mother, the One who held Thee in her arms and said: The Grace of Him Who has been born of me is spread through the world!
  Enlighten and direct my soul, my heart and my spirit. Guide the hands of Thine unworthy servant so that I may worthily and perfectly portray Thine Icon, that of Thy Mother, and all the Saints, for the glory, joy and adornment of Thy Holy Church.
  Forgive my sins and the sins of those who will venerate these icons and who, kneeling devoutly before them, give homage to those they represent.
  Protect them from all evil and instruct them with good counsel. This I ask through the intercession of Thy most Holy Mother, the Apostle Luke, and all the Saints. Amen.

...
Simply put:

Yes, Christians do still make Icons!


What is the process of removing an automatic excommunication?

There are indeed some sins which cause excommunication simply by being committed. Such excommunications are called latae sententiae excommunications (in Latin, more or less, "[excommunications] of a hidden sentence"). Acts which can incur such an excommunication include:

a physical assault against a bishop or the Pope;
procurement of an abortion;
attempting to celebrate the Eucharist, even though not an ordained priest.

The Catechism of the Catholic Church states:

Certain particularly grave sins incur excommunication, the most severe ecclesiastical penalty, which impedes the reception of the sacraments and the exercise of certain ecclesiastical acts, and for which absolution consequently cannot be granted, according to canon law, except by the Pope, the bishop of the place or priests authorized by them. In danger of death any priest, even if deprived of faculties for hearing confessions, can absolve from every sin and excommunication.

(paragraph 1463)
In particular, the Code of Canon Law provides:

Canon 1354 §3 If the Apostolic See has reserved the remission of a penalty to itself or to others, the reservation must be interpreted strictly.
Canon 1356 §1. The following can remit a ferendae sententiae or latae sententiae penalty established by a precept not issued by the Apostolic See:

the ordinary of the place where the offender is present;

Canon 1357 §1. ... a confessor can remit in the internal sacramental forum an undeclared latae sententiae censure of excommunication or interdict if it is burdensome for the penitent to remain in the state of grave sin during the time necessary for the competent superior to make provision.
§2. In granting the remission, the confessor is to impose on the penitent, under the penalty of reincidence, the obligation of making recourse within a month to the competent superior or to a priest endowed with the faculty and the obligation of obeying his mandates; in the meantime he is to impose a suitable penance and, insofar as it is demanded, reparation of any scandal and damage; however, recourse can also be made through the confessor, without mention of the name.

Here's what that means:

If a particular excommunication is specified as only being removable by the Pope (for example, if someone assaults the Pope), then officially the person needs to confess to the Pope. In practice, the Pope would probably depute the local bishop to hear the confession and remit the penalty.
In other cases, the local bishop himself (or a priest deputized by the bishop) would need to hear the penitent's confession.
In any case the confessor would impose an appropriate penance which would have to be observed.
After that, with sins forgiven, the person would be innocent and once more in good standing with the Church.

As far as whether a priest would absolve these mortal sins: Catholic priests are obligated to absolve from sin any penitent who seeks reconciliation and genuinely appears sincere, regardless of what their sins are:

If the confessor has no doubt about the disposition of the penitent, and the penitent seeks absolution, absolution is to be neither refused nor deferred.

(Code of Canon Law, Canon 980)


Looking for a resource discussing the Biblical principles on how to avoid a victim mentality

For some reason modern popular authors that deal with this subject are more of the autobiographical nature, such as Joni Eareckson Tada who has various books and lectures about taking joy in God's sovereignty, where she herself has been in a wheelchair for over thirty years.  Another person along the same line that I remember is Corrie ten Boom whose story of trying to love and evangelize people in Nazi concentration camp was published under the name'The Hiding Place'.  However, although these show great faith and lack of the victim mindset, they are not theological works but works of example.

As far as theological works go, there really are not many. Some might argue a famous work like 'The Christian in Complete Armour; A Treatise Of the Saints War against the Devil', By William Gernel is what you are looking for, and possibly it is. However, although many give this work very high reviews, I found it a little tiring to read and not really delving in the questions you are raising.  It might almost seem to veer slightly into a stoics approach to the 'warfare' of life.  It certainly leaves no room for a victim mentality, but still I would not say 'this is it'.

The only book I can think of that properly deals directly with the subject and is a classic, is Dr.  Martin-Lloyd Jones, 'Spiritual Depression: Its Causes and Cure'.  The title is a little misleading because although it's been about twenty years since I read it, I remember it would depress a lot of people.  It spends some time showing that the gospel really has all we need to rejoice in our sufferings.  I do not remember it being stoical, but at the same time self-pity is not spared at all in this deep theological work.


Quote like "Anyone can endure an insult but it takes a saint to pass through praise unharmed"

The quote comes from "THE LADDER OF DIVINE ASCENT" by St John Climacus.

http://www.prudencetrue.com/images/TheLadderofDivineAscent.pdf

On page 73

  People of high spirit bear offence nobly and gladly, but only holy people and saints can pass through praise without harm.


How common is intinction in the Catholic Church?

All Catholic churches I have attended have you line up to the front, you are given a communion (either placed in your hand or in your mouth), and on the way back to your seat is a wine bearer, who will offer you a sip from the cup. Unless you place the host in your mouth and then partake from the wine cup, it would not be possible to do this.

In other churches, as you have said, this can be common, but at a standard Catholic church it's not frowned upon as much as not possible following the standard setup.


Was Mary’s appearance to St. James the Greater in year 40 AD an apparition or bilocation?

This traditional event in the life of St. James the Apostle is called an apparition. It is most probable that the Virgin Mary was still alive at the time of this event, thus making it a bilocation.

  Venerable Maria of Agreda who was shown the life of the Blessed Virgin Mary in detail, writes in her Mystical City of God that St. James, brother of St. John, whom Our Lord called “Sons of Thunder”, had a special devotion to the Blessed Mother.

  Becoming the apostle of what today is Spain, Saint James was having a hard time evangelizing the northern region of Zaragoza. One night, as he prayed asking help for his plight, he suddenly beheld a great light in the midst of which he saw Our Lady surrounded by a multitude of angels.

  The interesting thing is that Mary was still living in Jerusalem at the time. But as queen of the Church, she was given to see all that concerned her Son’s work, and being shown the prayer of her devotee, had obtained from Jesus to help him in a special way. - The Marvelous Story of Our Lady of the Pillar

Many stories abound about this particular tradition. No Church Father speaks of it. and it is likewise not mentioned in the Scriptures, thus making it part of private revelation. No Catholic is under obligation to believe this legend, but many do, including myself.

  Several Christian saints, monks and Muslim sufis are said to have exhibited bilocation. Among the earliest is the apparition of Our Lady of the Pillar in the year 40. Other Christian figures said to have experienced bilocation include St. Alphonsus Marie De'Liguori (Founder of Redemptorist Congregation), St. Anthony of Padua, Ursula Micaela Morata, St. Gerard Majella, Charles of Mount Argus, Padre Pio,[13] St. Severus of Ravenna, St. Ambrose of Milan, María de Ágreda,[14] and St. Martin de Porres, María de León Bello y Delgado, as well as Pope Cyril VI of Alexandria. St Isidore the Laborer claimed to be praying or attending to Mass in Church while at the same time plowing in the fields. - Bilocation (Wikipedia)


Does agape contain affection for others whom the agape is directed to?
Short answer: Yes(it can, but it does not seem to be required to).

Discussion
Definition: Agape (Ancient Greek: ἀγάπη) is "love: the highest form of love, charity; the love of God for man and of man for God."

The Catholic teaching on Charity.

The catholic teaching is that agape/charity, is the greatest theological virtue.  From the Catechism:

  1822 Charity is the theological virtue by which we love God above all things for his own sake, and our neighbor as ourselves for the love of God.
  1823 Jesus makes charity the new commandment(John 13:34). By
  loving his own "to the end," (John 13:1) he makes manifest the
  Father's love which he receives. By loving one another, the disciples
  imitate the love of Jesus which they themselves receive.  Whence Jesus
  says: "As the Father has loved me, so have I loved you; abide in my
  love." and again: "This is my commandment, that you love one another
  as I have loved you." (John 15:9; 12)

  1824 Fruit of the Spirit and fullness of the Law, charity keeps the
  ommandments of God and his Christ: "Abide in my love. If you keep my
  commandments, you will abide in my love."
  1825 Christ died out of love for us, while we were still "enemies." The Lord asks us to love as he does, even our enemies, to make ourselves the neighbor of those farthest away, and to love children and the poor as Christ himself.

    The Apostle Paul has given an incomparable depiction of charity:
    "charity is patient and kind, charity is not jealous or boastful; it
    is not arrogant or rude. Charity does not insist on its own way; it is
    not irritable or resentful; it does not rejoice at wrong, but rejoices
    in the right. Charity bears all things, believes all things, hopes all
    things, endures all things." (1 Corinthians 13).

  1826 "If I . . . have not charity," says the Apostle, "I am nothing."
  Whatever my privilege, service, or even virtue, "if I . . . have not
  charity, I gain nothing."  Charity is superior to all the virtues. It
  is the first of the theological virtues: "So faith, hope, charity
  abide, these three. But the greatest of these is charity." (1
  Corithians, 13:13)

St Basil:

  If we turn away from evil out of fear of punishment, we are in the
  position of slaves. If we pursue the enticement of wages, . . . we
  resemble mercenaries. Finally if we obey for the sake of the good
  itself and out of love for him who commands . . . we are in the
  position of children.

St Augustine:

  Love is itself the fulfillment of all our works. There is the goal;
  that is why we run: we run toward it, and once we reach it, in it we
  shall find rest.

On an experiential note, in the past six years I have been involved with a lot of lay retreats that place heavy emphasis on the virtue of charity, and the principle of agape as the theme of the retreats.  Agape is alive and well, and being emphasized.


Where do I go to get official Catholic answers?

The bishop, or at least the Ordinary of the particular church, is the one who has ultimate pastoral care over all the people in his diocese:

In exercising the function of a pastor, a diocesan bishop is to show himself concerned for all the Christian faithful entrusted to his care, of whatever age, condition, or nationality they are, whether living in the territory or staying there temporarily; he is also to extend an apostolic spirit to those who are not able to make sufficient use of ordinary pastoral care because of the condition of their life and to those who no longer practice their religion.
A diocesan bishop ... is bound to propose and explain to the faithful the truths of the faith which are to be believed and applied to morals.

(Code of Canon Law, Canon 383 section 1 and Canon 386 section 1)
Because under typical circumstances a bishop simply doesn't have sufficient resources to provide adequate pastoral care to each and every person in the diocese, priests are appointed to care for those in particular areas, including to counsel them in matters such as those you describe.
Unfortunately, it does occasionally happen that priests do let their personal understandings get in the way of official Church teaching. If you are concerned that this may be happening, or if you just want to get more understanding of a particular issue, you may be able to get help from your diocese, which may have an office of evangelization or something similar (here, for example, is a link to the Evangelization Office of the Archdiocese of St. Louis).
The Catechism of the Catholic Church is also an important resource, of course; but it isn't primarily intended as a direct source for answering questions. Instead, it's intended as a reference for those who will pass on Catholic teaching to others:

This work is intended primarily for those responsible for catechesis: first of all the bishops, as teachers of the faith and pastors of the Church. It is offered to them as an instrument in fulfilling their responsibility of teaching the People of God. Through the bishops, it is addressed to redactors of catechisms, to priests, and to catechists. It will also be useful reading for all other Christian faithful.

(Catechism of the Catholic Church, paragraph 12)
In addition, there are many books and websites which will give Catholic answers to questions. Only a relative few, however, can be considered as containing content which is specifically concordant with the official Magisterium of the Church; so I will refrain from specifying even those which I personally have found most useful.


What is the biblical basis against euthanasia?

The biblical argument, I believe, is that as Christians we all must have faith, and just as Christ was able to wake the "dead girl who was sleeping" (Mat. 9) so we should have faith that Christ can heal. Things get a little bit more ambiguous when you're talking about brain death, though — we are able to keep body parts alive long after the original owner has clearly passed on (a transplanted heart will often out-survive its original owner for years), but committing suicide, in any circumstance, has been considered a profound lack of hope.

A non-biblical argument (put forth by Augustine) was that, more or less, "dying" had no substantive meaning — I could have cancer and be told that i have two weeks to live. Clearly I would be "dying" in that case, but I could then make a full recovery tomorrow. On the other hand, I could be perfectly healthy and then decide to go for a walk and get hit by a bus. In this case, the person who seemed healthy was the one who was really dying and the person who seemed sick was not dying at all!


What is the origin of the story of the angel Raphael and the boy Tobias?

This is a quote from the Book of Tobit, which is recognized as part of the deuterocanon by the Catholic, Eastern Orthodox, and Oriental Orthodox churches. The book tells of a father, Tobit, and his son, Tobias or Tobiah. Tobias is led by the angel Raphael to find a wife, and returns with medicine to cure his father's blindness. As Raphael prepares to go, he reveals that he is an angel, but before he does so, he gives them advice, including this sentence:

Do good, and evil will not overtake you.
Tobit 12:7b (New American Bible, Revised Edition)

Arthur Miller was Jewish, and the Book of Tobit may have been familiar to him, though it is not in the current Jewish canon. Puritans may also have been familiar with the book, though they follow the common 66-book Old Testament canon used by most Protestants.


How do those who believe or ascribe to Hypostatic Union interpret Philippians 2:7?

"Does he mean God "puts aside" his divine attributes when he takes the form of Jesus?"

No.

This is much related to the glorification of Jesus in John 17. As opposed to Paul's alluding of the putting off of glory when he says that Christ "emptied Himself, taking the form of a slave, becoming in the likeness of men," and was "found in fashion as a man"(Phil. 2:7-8).

  And now, glorify Me along with Yourself, Father, with the glory which
  I had with You  before the world was. (17:5) I have glorified You on
  earth, finishing the work which You have given Me to do. (17:4)

Glory is ontologically the mode of the Divine Being in which God manifests Himself. In the Old Testament God's people could, at most, look upon the glory. The prophets introduced the notion that in the future the glory of God will be an enduring reality that will fill the earth, but even then the glory is only with God as the expression of His manifest Being dwelling among humankind.

Ramm, in his book, "Them He Glorified: A Systematic Study of the Doctrine of Glorification", observes that "it is a kind of totality of qualities which make up His divine power," and he concludes:

  the glory of God is not...a particularized attribute like the wisdom
  of God but an attribute of the total nature of God, virtually an
  attribute of the attributes”.

It is in the person of Christ that the ontological and eschatological aspects of glory
meet, for in Him we behold the God of glory and in Him we shall be glorified. The
Gospel of John is especially concerned with this theme. In fact, we can easily define
the sub-theme of this Gospel as the incarnated Glory of God on the path to bringing
His believers into the divine glory through death and resurrection.

The opening words of both the Gospel of John and the Epistle to the Hebrews refer to the Son in His eternal existence as glory. The Son eternally exists as the expression and glory of the Father, and the distinct function of the Son is to make the Father manifest.
The son is identified as the glory of the Only begotten of the Father (1:14). By this we should understand the hypostatic nature of glory in the Godhead—glory not merely as an attribute of God but as the unique hypostasis of the Logos as the expression of God.

We cannot construe this emptying - the so called kenosis (from the Greek verb for emptied here) as a putting off of His divinity in toto; otherwise, He would not have been God at all during His earthly life, and His death on the cross would have lost its redemptive efficacy.

Rather, in simple terms, after the Lord became flesh, He expressed God to some extent, but the glory of God, the glory of all His attributes, was hidden in His flesh. The glory of all God's fullness was covered, being clothed by His flesh [i.e. emptying]. It was revealed once on the Mount of Transfiguration.

Thus, Jesus told His disciples that He had to die and be resurrected. This is liken to the glory and beauty of the flower that are hidden in the seed are released by its death [c.f. John 12:24].

The "colorful beauty and the rich glory" are released through death and resurrection. This is glorification. After the Lord told His disciples about His impending death and resurrection, He offered a prayer to the Father, praying that the Father would glorify the Son that all the glory hidden within Him might be manifested through His death and resurrection. To glorify the Son means to bring forth all the glory hidden within His flesh, to release by means of death and resurrection all the divine glory concealed within the flesh of the Son.

Typologically, just as God’s shekinah glory was concealed within the tabernacle, His divine element was confined in His humanity.


From a Catholic perspective, what do family relationships look like in heaven?

What is the beatitude of heaven?

Catechism of the Catholic Church 1027 answers:

  This mystery of blessed communion with God and all who are in Christ
  is beyond all understanding and description. Scripture speaks of it in
  images: life, light, peace, wedding feast, wine of the kingdom, the
  Father's house, the heavenly Jerusalem, paradise: "no eye has seen,
  nor ear heard, nor the heart of man conceived, what God has prepared
  for those who love him."1

  1. 1 Cor
  2:9.

Thus it is the blessed ( = supremely happy) 1) communion with God and 2) communion with all who are in Christ.

Catholic theology expands on this by calling the former "Essential beatitude" and the latter "Accidental beatitude". And part of the accidental beatitude is:

  They delight greatly in the company of Christ, the angels, and the
  saints, and in the reunion with so many who were dear to them on
  earth2.

2. cf. Accidental beatitude in Heaven | New Advent

They way I see it, it will be coming home to one great big family reunion, wonderful and beyond all telling. cf. Matt 8:11:

  11 I tell you, many will come from east and west and sit at table with
  Abraham, Isaac, and Jacob in the kingdom of heaven[.]


What is the basis for the belief that there were twelve Magi?

The basis for the belief that there were twelve Magi (or more) is a very obscure, apocryphal manuscript called The Revelation of The Magi that was discovered in a Turkish monastery in the 18th century.

This Syriac manuscript can be found in the Zuqnin Chronicle (1) and the Opus Imperfectum in Matthaeum (2) and currently resides with The Vatican (Biblioteca apostolica, syr. 162).

Brent Landau has published a dissertation on the subject: The Sages and the Star-Child: An Introduction to the Revelation of the Magi, An Ancient Christian Apocryphon

From Landau's summary and introduction to the text:

  It is also worth mentioning that the list of the names of the twelve
  Magi and their fathers in Rev. Magi 2:3 is found in works by a number
  of other Syriac writers. However, the names from this list are never
  referenced again in the narrative of Rev. Magi, and they are so
  widespread throughout Syriac literature that they are likely not to
  have originated in Rev. Magi, but instead were added to the text at
  some point after its creation. Indeed, there are hints scattered
  throughout Rev. Magi that the Magi constituted a group much larger
  than twelve: the text uses the Syriac term mashritha to describe them
  on several occasions, a word used in the Syriac NT to translate the
  Greek parembolos, used almost exclusively for large assemblies of
  people (e.g., Acts 21:34; Heb 11:34, 13:11). (4, emphases added)


Who were the 40 men who wrote the Bible?

The 40-author count is not unique to Jehovah's Witnesses, so I'll provide the names of the forty commonly-accepted authors. Note that the authorship of some books (Hebrews, for example) is debated, so the number and names vary slightly depending on which scholars you ask.

Moses
Joshua
Samuel
Ezra
Jeremiah
Nehemiah
Mordecai
Job
Sons of Korah
Asaph
Heman
Ethan
Hezekiah
David
Solomon
Agur
Lemuel
Isaiah
Ezekiel
Daniel
Hosea
Joel
Amos
Obadiah
Jonah
Micah
Nahum
Habakkuk
Zephaniah
Haggai
Zechariah
Malachi
Matthew
John
Mark
Luke
Paul
James
Peter
Jude

See When was the Bible written and who wrote it? for a matchup of names to books, along with approximate dates of authorship.


What are the positions relative to each other in the Catholic church's clerical hierarchy?

The Catholic Church has the Pope as its head. The basic structure underneath the Pope consists of a number of bishops, each of whom is assisted by a number of priests. Priests, in turn, are assisted by deacons (who are ordained, just as priests are, but who cannot "advance in the rankings" as priests can).

However, there are a number of variances from this, and a lot of details to go into. Are you sitting comfortably? Then I'll begin ...

To understand the Church's governing structure, keep in mind that the Catholic Church is divided up into what are called particular churches. Most of the particular churches govern a specific area (usually called a diocese or eparchy) and are aggregated into a number of what are called a sui iuris ("in its own right") churches; the largest of these is the Latin Church. The various particular churches are each overseen by a bishop. (Some of the heads of the sui iuris churches called the "Eastern Catholic Churches" are given the title Patriarch. For example, the head of the Syriac Catholic Church, who is a bishop, has the title Patriarch of Antioch. The rest of the heads of the particular churches making up the various Eastern Churches are generally called eparchs.) Bishops are appointed by the Pope, with the advice of a dicastery (a standing department in the Vatican) called the Congregation for Bishops; they are ordained priests who are consecrated as bishops either by the Pope himself or by other bishops.

There are some regions of the world which have a fair amount of Catholics, but not enough to bear organization into a diocese led by a bishop. The Pope may organize these into an apostolic prefecture or the slightly more advanced apostolic vicariate, led respectively by a prefect apostolic or the higher-ranking vicar apostolic, an experienced priest—but not a bishop—who is entrusted with the spiritual care of the Catholics in the area. The Eastern Catholic Church equivalent of a vicar apostolic is an exarch.

Since an individual bishop isn't able to minister directly to perhaps tens or hundreds of thousands of people in his church, the Church allows him to delegate certain functions to priests, who (usually) are assigned to local areas, known as parishes, within the particular church. Some of these priests have historically been rewarded for their work by being designated Monsignor by their bishop, a title which doesn't carry any particular increase in rank. Often, as well, priests are assisted in their work by deacons, either transitional deacons who are seminary students studying for the priesthood, or permanent deacons who are ordained to the Order of Deacons, but will not advance to the priesthood.

Since some geographical areas are inevitably more economically and politically important, or simply more highly populated, than others, the dioceses or eparchies—the "particular churches governing a specific area"—are often grouped together into relatively small groups called ecclesiastical provinces, often coincident with political divisions such as states or provinces. Generally, the diocese covering the most important region of an ecclesiastical province will be ranked as an archdiocese (or archeparchy), usually what's called a metropolitan archdiocese, and will be governed by an archbishop—generally also eligible to be called a metropolitan bishop. For example, within the U.S. state of New Jersey, there are five dioceses: the Diocese of Camden, the Diocese of Trenton, the Diocese of Metuchen, the Diocese of Paterson, and the Archdiocese of Newark (the largest of the four cities, and the closest to New York City). A metropolitan archbishop doesn't rank above a bishop, but does have certain types of responsibilities and oversight over the other bishops in the province, who may be known as suffragan bishops.

Many dioceses, as well, have multiple bishops assigned to them—a "main" bishop, and one or more auxiliary bishops. (All the auxiliary bishops are suffragan bishops as well.)

A chorbishop, as I only just now found out, is a rank just below bishop in some Eastern Churches; these men have many of the duties and powers of a bishop, but not all. They are therefore ordained priests as well, but not bishops.

Some bishops are chosen by the Pope to join the group of clergy responsible for the selection of future popes. These are called cardinals. There are three ranks of cardinals: cardinal deacon, cardinal priest, and cardinal bishop. The cardinal bishops are the most senior leaders in the Church after the Pope, and cardinals as a whole rank above bishops. (Not all cardinals actually can select the next pope; cardinals are in office for life, but cannot vote once they reach 80 years old.)

As well as all these (as if they weren't enough!), there are also various religious institutes. These are often governed by a superior (more or less a supreme leader), who may have any of a number of titles. The Augustinians call their superior the Prior General; the Jesuits, the Superior General. These are equivalent to bishops.

Many religious orders are governed locally, organized into abbeys, convents, or priories. They are subject directly to their leader, who is subject (more or less) only to the Pope, and is thus pretty much equivalent to a bishop. This local leader may be an abbot or abbess, a mother superior, or a prior.

There are even more titles that I haven't gone into, but they're generally reserved for specific individuals (such as the senior cardinal-priest in the College of Cardinals, who is known as the protopriest). I won't go into all of those; this post is long enough already.

You mention a few titles which are not (or are no longer) titles of clerical personnel:

Exorcist: Before Vatican II, this was an official position ranking below a priest. Now, it's a function that's exercised by a very few priests—usually only one per diocese or eparchy. The exorcist is responsible for conducting exorcisms. No, not always that kind of exorcism ... but sometimes.
Lector: Again, before Vatican II, this was an official position ranking below a priest, and like exorcist, leading up to ordination. Nowadays, it's a position often assigned to a lay minister.


What do the words Catholic and Protestant mean?

Catholic can have a few different meanings. It could refer to:

The Roman Catholic Church
Eastern Catholic churches in communion with the Pope
The universal church

The Nicene creed which is used in a lot of Christian churches that have broken away from the Roman Catholic church still say that they "believe in one holy and catholic". (note the lower case letter 'c' in catholic.) Catholic in this context does not refer to the Roman Catholic church, but to a holy Universal church.

Protestant refers to churches and/or individuals who call themselves Christians, but disagree with the Roman Catholic Church.

  The word 'Protestant' derives from the protests made by German princes
  at the Second Diet of Speyer in 1529. The Diet voted to end the
  toleration of those who followed the teachings of Martin Luther within
  Germany (source)

They generally follow the teachings of leaders of the reformation (e.g. Luther). They are called Protestant because they protested the teachings of the Roman Catholic church. The original Protestant movement came from within the Roman Catholic church. The teachings of the protestant movement disagree with Roman Catholicism and a number of the reformers were put to death because they taught these things.

Do these terms still fit today? I don't think this can be answered objectively, but I feel that they are still a correct fit. Mainstream Protestant denominations still hold to the Five Soli. From my understanding, these beliefs do not fit with the Roman Catholic church teachings and so Protestants are still protesting the Catholic Church