This is the main issue. iTunes recognizes the device but just refuses to sync.
The iPod may be set to Manually manage music and videos, or it was associated with a different iTunes library (perhaps your old iTunes library if different from current iTunes library).
If set to Manually manage, Enable disk use is also enabled. That means iPod's "disk" is mounted continuously by the system, so its screen shows "Do not disconnect" while connected, until you Eject it in iTunes. AND clicking Sync does not sync any music, if set to Manually manage. So, it fits your description (without anything being "wrong"). Also, when you do a Restore ("wiping the iPod"), it is set by default to Manually manage. You must set it to Sync Music.
If you want it to automatically sync your music library, go to iPod's Summary settings screen. This screenshot shows my iTunes window (upper-left corner), with iPod's device button circled in red.
Click that button to show iPod's Summary settings screen in iTunes window. Along the left side, in the sidebar, under Settings, click Music. iPod's Music settings screen appears to the right. Checkmark the box for Sync Music
NOTE: For anyone else reading this, any songs on iPod that are not in your iTunes library will be lost.
This turns ON automatic syncing. Chose option to sync Entire music library (if it fits on iPod). Click Apply to sync. Going forward, iTunes keeps iPod "in sync" with changes to your music library, whenever you connect iPod (or click Sync if already connected).
