Can you delete an app and get it back and still be logged in?

Some apps maintain data on servers and you have to log into those apps. GameCenter apps are one class of those.

Thus, if yo use the same login your data will be maintained/restored.

It all depends upon the app.

If you delete an app it should sign you out and delete all app data as well. Are you experiencing a problem? If so, please describe it in as much detail as possible.