If I replace an iPod Classic battery myself, will the music be preserved?
The iPod classic has no NVRAM/PRAM or settings that degrade when battery power is disconnected. Unless you want to roll the risk that opening the device causes irreparable harm into the definition of "replacing the battery" - you are safe to go. Just like you can pull a hard drive and place it in a new computer - same with the drives in a classic iPod - everything important is on the drive.

I would put this in the class of putting a spare tire on a car. Even if you've never done it before, you get out the manual and most people can do this themselves - especially if you are not in a rush. Likewise, those that really shouldn't attempt the operation generally have a good intuition that they are not up to the task.

I say, dive in and replace your battery once you've read up on which connectors are fragile for your specific model. Cracking the case is by far the hardest part of the surgery.