The ConstantsThe standard conception is that these physical constants are a fundamental feature of reality. But if you take a look at how physicists works, you get a very different picture.
The truth is that physicists really don't like constants. If a theory ends up having too many constants (or "fudge factors" as some scientists call them), then it begins to look suspicious. The main reason is that each constant is an unanswered question. There's generally a deeper reason for the value of a constant. For instance, the speed of sound in air seems constant, but it turns out to be related to the inertia of the air, so we can calculate it from other values. Scientific progress is gradually supplying answers to these unanswered questions, so positing God as the answer is a cut-and-dry God of the gaps argument.
But there's more to it than just the unsatisfied question of "why?". A physical theory with a lot of constants is like a house for sale that's filled with overpowering air fresheners: there's a lot of room for error..."overfitting" and sometimes "data dredging", in this case. With enough constants, any model can be made to fit the existing data, but theories with more constants have much less chance of predicting new data, and therefore are likely to be wrong as an abstract model. It's not always a case of foul play, but the values of any constants will be fit to existing data, which means that more constants couple a model to specific measurements and steer away from a general solution. The building blocks of our physical theories seem "undeniably real", but that's more because we've been taught about them for so much of our lives than because it's so obvious.
All this muddles the question of what the constants actually represent, and what it means to "tune" them.
How Things Might Have BeenEven if these values can be freely changed and the universe could be drastically altered, what then? The fine-tuned constants argument, like all design arguments, paints our environment as somehow "special". In this case, it's usually claimed that life could not exist in a universe where such-and-such value were different by one part in some-odd million billion billion billion.
But the same "scientists" have until now been telling us that evolution is impossible, and therefore that life should not exist in our universe. Sure, an altered universe would be different, but the salient question is "could (intelligent) life exist?". No scientist today is qualified to start with a handful of values, extrapolate out to an entire universe, and then determine that intelligent, self-replicating life won't ever exist anywhere within that universe. And certainly no scientist is qualified to do that same thing for every possible universe to determine that all or even most would be uninhabitable.
Answering the QuestionsTheistic arguments take an attitude of demanding an answer for a given question. They often take the form "how else can you explain it?". That's not always a terrible approach, but it's certainly something to take in moderation, since easy answers are often wrong. It also entails the assumption that we already have all the requisite knowledge, which is especially unlikely when we're talking about metaphysics.
The common response is to take these arguments at face value and try to give an alternative explanation. To that end, many people suggest anthropic reasoning to explain why we observe such "special" conditions in our universe. I believe the anthropic principle is an excellent answer to the question, but based on the above arguments I don't believe the question itself is valid. The anthropic principle is such a neat, unconventional idea that it's easy to forget to ask if it's really necessary in this case.
Finally, it's unrealistic to expect scientific inquiry to ever finish its job and answer every significant question. The answers to questions about the "fundamental constants" will probably be incredible, but the questions themselves aren't all that special or compelling. There are volumes of unanswered questions; what makes this question of fundamental constants stand out is mostly that it's easy for laymen to speculate about.
The bottom line is that the argument from fine-tuned constants is a lot of things, but it isn't science.