You can try to diagnose the problem with the diagnostic mode.
To enter the diagnostic mode, press the folowing key when the apple logo appears after a reset :
iPod 1G, 2G et 3G, press the keys for 5 seconds :
iPod 4G, 5G, mini et nano, press the keys for 5 seconds :
In the diagnotic mode, you can navigate in the menus with the back/next keys and launch a bunch of test.
I have an alternative take on this problem:
I have just bought a Classic 160 Gb at Amazon few weeks ago and I have been facing the same issues as you.
I do not use Mac OS X, nor iTunes to sync. I run Linux and I use Media Monkey on a Windows VirtualBox Machine.
During the past two weeks I have tried everything that I could find. iTunes have not even been able to read my entire collection (strange, right?)
My findings
I have just restored the iPod firmware a few minutes ago. Started MediaMonkey and sync'd 10 songs from an ordinary random artist. Disconnected and the songs were there, nice. So the iPod is actually working. So far, so good, I have figured that out before.
However, in the meantime, while I was checking my library, I recalled that I have some songs from Japanese bands with Japanese characters in their names. Historically, this would be very error prone. So I did gave it a try, and sync'd 64 songs from that band. Disconnected the iPod and the songs were all gone.
So apparently the 2.0.4 firmware does not handle these characters very well. Check your library, make smaller tests. I am about to restore the iPod again and will try to sync some more files with normal ASCII characters. Will update you.
My old, now dead (waiting for a new disk), iPod, running 1.0.2 was able to hold those files nicely.
Best Answer
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.