Is it possible for a restored iPod touch to recover former apps that had been installed on an iPod touch?
Is it possible for a deleted app to be re installed containing the same information from its previous use, and also still be signed in?
Yes, just restore from an iCloud or iTunes backup.
If you're talking about forensics / recovering deleted files, that won't happen unless you have a lot of time and/or a lot of money. (And you probably won't get anything back.)