The JFBConnect Authentication plugin was unpublished, which was preventing logging in.
Looking at your configuration, you do have some required fields, but they aren't showing up on the registration form -and- more importantly, even when not set, they aren't causing CB to choke when saving the user (which is what normally happens). I'm not sure why this isn't happening for you, but it's actually a good thing. The required fields I see are all related to the user's current location.
The difference between Full Joomla User and Facebook Only is that with Full Joomla, the user gets to select their username, set a password, and (possibly) fill out additional information for their profile. It's more personalized, but adds a few steps to registration. For Facebook Only, JFBConnect automatically generates a username, and the user can then be redirected to their profile to optionally fill it out, if you want. This is faster to get users in (which is great for a sales site) but not always great for more of a community site. Hope that helps explain.
Everything seems configured well on your site. Again, still not sure why the required fields aren't showing up, and aren't causing CB issues when we try to save. It's a good thing though, but do watch out for any users who can't login. If you find ones with issues, check the user in CB, and a good indicator that CB had issues saving them is that their account will be Enabled but not Confirmed or Approved. If you start seeing that, let us know.
Hopefully, your redirect issues will go away now and all will be peaceful in kayak-land