Apple Music contains DRM and cannot be used on an iPod classic.
If iTunes won't display the device at all then Windows users should try iOS device not showing in iTunes for Windows. Despite the title it appears that it may help with regular old school iPods too. Mac users are probably best off attempting to reinstall iTunes. I don't offhand know of other useful troubleshooting steps for Macs but no doubt they exist.
iTunes is designed to sync devices with a single library. If you change computers without migrating the library properly, or the library's identity is reset in some way, you may find that your sync selections are missing and that simply clicking sync doesn't give the results you expect.
The Corrupt iPod Classic tip covers a number of aspects, but principally discusses how you might assess the health of the device's hard drive. If it is no longer reliable then syncing and playback are going to have issues too. If all hard to read sectors (pendings) can be cleared, and new ones don't appear after syncing then you may get more use out of it. Also included is a staged transfer method that can help when reloading content, and links to media recovery options if there is content on the device that isn't in your library.
I'd be almost certain that your iPod is running the latest firmware.
tt2