Hi Jack,
Your question implies a basic misunderstanding of this site. You ask, "When you guys replace batteries..." but the fact is, iFixit does not do repairs; the whole point of this site is to enable you to do your own repairs.
But to answer your question, no you normally would not lose your music when the battery is replaced. Whether it's a physical hard drive or solid state memory, both are non-volatile, meaning they don't lose their contents when power is removed.
Virtually all of the people who frequent this site such as myself are volunteers who try and help out with answering questions and making our experience available to help those who need assistance. So you can ask for advice from users, but there's not going to be any kind of 'official' stance or position on repairs or parts from iFixit itself.
As far as my own suggestions, if they're working well then I'd say leave well enough alone unless you're having specific issues such as short battery life. A straight-forward battery replacement is most likely all it needs in that case. You can always look at new cases if yours are scuffed and dented if you want to try and restore it cosmetically, and for the ones with hard drives there are solid state memory conversions available; I personally have a 20 Gb 4th Gen iPod that I've installed adapters so it's now running on a 128 Gb micro SD card.