The FirebaseAuthClient should automatically session the user and keep the user logged in, but it is possible that there is something going on preventing that. Ping me off-list with a link to your app, or a sample of the auth. code that you're using? I'm happy to investigate.
2. How can I update a user's information?
Keep in mind that Simple Login is only authenticating the user using the authentication method that you specified (email / pass, Facebook, Twitter, etc.) as a convenience, but is not storing any data in your Firebase for you. This is intended to keep your Firebase data as flexible as possible, and allow you to store data however you wish. That means that after authenticating a user, you should set / update that user's data in Firebase. However, you cannot change the user's social data (Facebook, Twitter, GitHub, Persona, etc.) data via this API, and for e-mail / password login, we're only securely storing the user's credential for you, so there shouldn't be anything to change.
3. Is there detailed documentation for FirebaseAuthClient and how the user objects work? I looked high and low and couldn't find anything other than the overview page.
Check out the specific page for the login method that you're using to see the list of attributes that are guaranteed to be included in the user object. As mentioned above, these are immutable, and capture a snapshot of the user's information at the time of authentication.