Your question can be rewritten in short form:
- iTunes: How to move your music to a new computer
- Where are my iOS backups stored? - expand the section About Syncing and Backups and you'll see
iTunes places the backup files in the following places:
- Mac: ~/Library/Application Support/MobileSync/Backup/
- Windows XP: \Documents and Settings(username)\Application Data\Apple Computer\MobileSync\Backup\
- Windows Vista and Windows 7: \Users(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\
I linked to the official articles since things do change and cutting and pasting the steps today might not be right if things change in a month or two. Don't worry - the article about music moves everything except the iOS device backups. If your device is working, there's no need to move backups unless you want to keep older ones or save time the first time you sync with the new iTunes.
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
You are doing everything right. Sometimes an error on the other end (apple's servers) can look like a local problem to iTunes. Most of the time, what you did would have fixed this.
Have you considered getting a hand from the billing / account people that work for Apple? Start with the Account & Billing section on the left...
Your computer needs the OK from Apple's on line servers to fix the errors. This is an account issue and you don't have to pay for this support (but may have to start by emailing apple if no one is free when you ask for help).
You could try authorizing your account on any iOS device or another computer to rule out problems with your online account as well.
If you get errors on your computer after you re-authorize, you might have to go through this article on iTunes repeatedly prompts to authorize computer to fix things. If the iPod is giving the error, be sure to restore the iPod (which will wipe everything that's not backed up) to get the new authorization to take on the iPod.