I just tested on your site and was able to:
* Register with Facebook - My profile was imported as expected.
* Logout from the site.
* Log back in with Facebook.
This was all in the Chrome browser, for reference.
The 'Updating details of this app' may have resolved itself. Facebook periodically adjusts settings on apps and, during that brief period, users are unable to interact with them. Obviously keep an eye on it.
As for the Primary Key issue, if that happens every time you log out, I'd recommend setting the "Debug System" option in Joomla 'Yes' and set 'Error Reporting' to 'Maximum'. Then, logout again. That should provider more details as to the database table that is throwing that error, which should help narrow it down. I don't believe it will be from JFBConnect, but if it is, we'll gladly help diagnose and fix that issue as well.
Finally, we released JFBConnect v9.0 (final, not release candidate) yesterday along with Joomla 4's launch. It's still fully backward compatible with Joomla 3, and will be for some time. There aren't any major updates from the RC release and nothing that should affect the issues above, but always nice to be on the latest if you want to upgrade.
Thanks,
Alex