Hmm..Here's a few options:
~Hopefully you can see your Ipod under My Computer.
1. Enable the viewing of hidden files, then browse to (let's call it I:/ drive.)
I:/Your
Ipod_Name/iPodControl/Music
Copy the whole music folder to your desktop or somewhere safe. That folder contains all your songs, though they will have names such as HGB3.mp3.
(Enable Hidden Files<Under Advanced View>: Control Panel>Appearance and Themes>Folder Options>View Tab>Show Hidden Files and Folders)
~If it doesn't appear under My Computer or if it does.
2. Use a third-party software program to get the songs from your Ipod to a folder on your computer. This will get the songs from your Ipod with the correct artists and names. Not sure if posting links to third-party software is 'legal', but here it goes:
http://download.cnet.com/iDump/3000-2169_4-10407429.html
I use this one to keep a backup of my songs from time to time. It's best to keep the backup on an external drive just in-case you HD dies.
If your still not able to get a backup of your songs, well that's all I have right now. Maybe you can wait for other replies, or just go ahead and format it(but you lose your songs).
Now to restore:
Let's see why it isn't appearing under iTunes.
Open up iTunes with your ipod connected, ignore anything that may pop up. Click the tab called 'Help' and then 'Run Diagnostics...'. Check only the two last boxes:
-Iphone/Ipod connectivity tests
-Iphone/Ipod touch sync tests
Hit next, again make sure your ipod is connected...next one more time.
You should pass all tests for connectivity.
Now sync, which you should fail.
Apple wrote:
This feature tests iPod and iPhone connectivity on a Windows PC running iTunes. Except for syncing, the diagnostic tests in iTunes do not test the hardware or software on the iPod/iPhone.
If any problems occur under the connectivity tests, post here.
If your up to losing your songs, your ipod is treated like a volume so right-click > format, should format it, deleting everything. When connected to iTunes, it should present the option to restore.
^I can say this, because I've done it, but I don't know if the outcome will be the same for you.