Let's see...
"Hey, has anyone been able to play XM or IT music files without the use of FMOD or MikModCE?"
No, I was never able to get either to work in a PocketPC app. They DO work mind you, *I* just couldn't get them to work.
"The former costs money and the latter has staticy sound. Surely there must be a better solution?"
Always. Write your own player! Seriously though, if you can live with using MODs, check out PocketFrogPlus (assuming you want to use PocketFrog, since you posted in this forum!). It might not even be massively difficult to add XM or IT support to that, you have the full source in any case, it might be worth investigation.
"I'd much prefer to use XM or IT files instead of MODs for music because to convert MIDI music to MODs, ModPlug has to mess with the way the patterns are stored in rows."
Yeah, I had the same rude awakening. Very annoying. From everything I've read, there is no simple conversion method from MIDI to MOD, you'll have to re-track it at best.
"Should I even bother with background music? Do most games have it?"
I would. I think you'll find that most decent games do have music of some sort, whether it's background during the game or just on an intro screen or whatever. I think it might be fair to say that most games you get for PPC DO NOT have music in them, but the top games almost always do. Take that for what ut's worth.
"Is it possible to have smooth action on screen and play music at the same time?"
Absolutely! I'me able to get a solid 30 FPS with a MOD file playing using PocketFrogPlus on my Casio E-125 running at a stock 150MHz. I'm doing full-screen redraws of a 320x240 background with a couple of variously-sized sprite moving around. People are doing even better than that in most cases. It's absolutely possible, and not even particularly difficult it would seem. Just a question of what player and what file format is all.