Question: 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?

Answer: 
If you restore using a backup that is recent, depending on the app, associated information will get restored (i.e. most games will generally have your high scores) as long as the app uses the standard way of storing data on an iOS device (which optionally involves iCloud).

This is the information that is at `/var/mobile/Applications/{GUID}/Documents` generally. Some apps do not store certain items there for security or space (in the backup, such as large in app purchase content) reasons. The app itself will generally have a way to retrieve such data on first launch after you provide credentials.
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.)