Connect the iOS device to your computer and try to make a backup.
Then restore via iTunes. Place the iOS device in Recovery Mode if necessary to allow the restore.
If recovery mode does not work try DFU mode.
For how to restore:
To restore from backup see:
If you restore from iCloud backup the apps will be automatically downloaded. If you restore from iTunes backup the apps and music have to be in the iTunes library since synced media like apps and music are not included in the backup of the iOS device that iTunes makes.
You can redownload iTunes purchases by:
Yes, I think you did something incorrectly. Did the iPod (as shown in the status bar in iTunes) successfully go through all the restoring steps?
I would try DFU mode and then restore next.
Correct, no reason to backup. Just observe to see if all states of the restore complete. Sometimes the firmware download does not complete and times out and the restore really never starts. This leaves the iPod i the same condition (asks for screen-lock passcode).
You are not talking about the Restrictions passcode, right?