ipod classic backup

As noted there is no backup mechanism for regular iPods. All your content should be in your iTunes library ready to restore to the device should it be needed. If that isn't the case see Recover media from an iPod - Apple Community.

tt2

You can't back up an iPod Classic to iTunes on a computer. You might be able to use some kind of third party app.

Yes, that’s indeed possible. By using an application called “PodTrans” you can export the music of your iPod. That exported music can be drag&dropped in iTunes.