Exodus 15 is a tough one to reply to for numerous reasons. First of all, this is a song that the people are singing, so one could simply dismiss the entire thing as unimportant from a theological standpoint. Second of all, the chapter is largely a review of what we have just seen in the previous chapters, so it's going to be hard sorting out what's a repetitious issue and what is truly new. Thirdly, as this is by form a poem, there may be some use of poetic language here that clouds the meaning. Why does all this make it hard to give responses to? After all, all of those are excellent excuses to not have to respond at all, case closed. Well, I think that choosing any of those approaches that would be an easy out that I don't think would be fully satisfactory to most skeptics. Those issues will come up, and I think they are valid excuses, but I don't want to gloss over a difficult passage too simply.
So, the Israelites are singing a song of praise to God because he just killed a bunch of Egyptians. Sounds twisted, no doubt, but I think an important factor to remember is that said Egyptians were in the process of trying to hunt them down and kill them. I think we can forgive them a little rejoicing in such a case. Maybe all readers won't agree, but when you've been enslaved as a nation for 400 years, and your slave masters try to kill you as you make your escape, which you made with permission, you might be a little more than relieved to see them die.
Is God a "man of war"? My answer (and did you expect a better one?) is: it depends. God definitely has a preference for peace over war. Throughout the Bible, I see God as trying to get people to get along with each other, but it's part of His nature that He views it as far more important that they get along with Him specifically. War is an odd thing, though. In many ways, the rules change in the midst of war. For instance, it's generally accepted that if you shoot a man just because you don't like him, you're a murderer, but if you shoot a man because you're a soldier and he's an enemy combatant, you're a hero. And rules change on levels that are beyond the personal; borders get redrawn, international trade breaks down, and history is written by the winning side in the end. I think it may be the case that war makes the rules change all the way up to God's level, but it may also be that God's rules are different to begin with, and if so, I still think that's acceptable. (See first paragraph here, and entire post here on the parent-like relationship God has towards us.)
Most of the talk of violence in this chapter boils down to a single subject, that being that God is known for perpetrating acts of violence upon people. I think the answer to this is pretty much what I've already given here; the reasons that the Israelites have to rejoice are the same reasons God has to do what He does. As a side note, while the Bible doesn't point out the absurdities of many of the described actions of God other than the "nostrils" reference, it should be noted that this chapter is poetic through verse 21, and much of the language is taking poetic license as a result. God doesn't have arms, hands, or nostrils, but apparently it was thought better to visualize Him with them for purposes of retelling the story.
Also, note that a careful reading of verse 15:9 should indicate that this is Pharaoh's "lust", not God's. While God at times does violent things, I don't believe he "lusts" for violence. The issue of the number of Gods was addressed by me in full in a previous post.
Perhaps because this is the first mention of dancing in the Bible, the SAB takes a moment to ask the question, "Is dancing a sin?" While there are certainly some Christian denominations that express the sentiment that it is, I don't think you'll find any clear condemnations of it in the Bible itself. The three verses quoted by the SAB to indicate that dancing is a sin are actually pretty easy to explain. Exodus 32:19 is a verse in which Moses sees dancing and gets angry, yes, but in context, dancing is not the only thing happening. Chiefly, Moses is angry about the calf, but the dancing in this case may be akin to the issue of the Matthew verse. In Matthew 14:6-8, we see someone dancing, and while the Bible does not specifically point out the evil of this young woman, the SAB is right that there is something inherently wrong with the dancing here. Essentially, most scholars are of the opinion that this girl is doing a striptease for her stepfather. As for Galatians 5:19-21, dancing is not mentioned specifically, so I'm not sure why it's there at all.
Other than a parting thought which might be covered under the violence umbrella above as biological violence, the SAB finishes the chapter only noting the apparent absurdity of a tree being put in some water to make it palatable. I'm not sure why this is absurd. It may have been a miracle, but it might also have been a perfectly natural effect. Whatever tree this is, we do not know, nor what the effect was. Perhaps the water was okay, but just tasted bad, and this tree had a sweet taste to it, or even some sort of fruit that they made a punch out of, who knows? Anyway, I'm not sure what the real issue is here.