Thanks for the kind words regarding our support. Yes, we think support is one of the most critical parts of any extension.
As for your issues, regarding logging in, when you say "When the user tries to log back in", are you saying just coming back to the page when they're already logged into Facebook, or actively clicking the "Login Using Facebook" button? I can envision the redirect not working if you're expecting it to on automatic login, but just need some more info to narrow it down.
With logging out, if you have the "automatic login" set, if you just try to log out of Joomla, on the next page load, it would probably try to log you back in.. causing the issue your seeing. Also, if you were to refresh the page, would it then show you as logged in (similar to the refresh required above)? I'm not sure why the module shows you logged out though, and that is something we'll investigate.
I know this isn't very helpful. We originally had the auto-login feature back around version 2.2, but removed it in a later version for similar issues. When we sat down to rewrite the component for 3.0, we architected the component much better to be able to handle auto-login, but it's obviously still a little rough around the edges. We'll get it going again though.
We'll do some more investigation on our end, and if you could clarify the few questions above, that'd help a ton.