Florian,
In the JFBCLogin module, you can simply disable the "Allow Reconnections" option to prevent users from changing their mappings altogether. Its generally not a used feature, and we'd just recommend you disable it.
For the session database I mentioned, you'd need to run a check on your database tables. You can do so by running CHECK TABLE xxx_session. Also, components like Akeeba's Admin Tools have the ability to check and repair tables automatically.
Changing the session has the effect of increasing your database size slightly, since more rows (1 for each visitor on your site) will stay around longer. Additionally, if user's don't use the logout button, they will stay logged in for the full session length. If your site is frequented by users in internet cafes, keeping this smaller is better.
Finally, the 2 sites thing is extremely strange, and may be part of other issues. It sounds like you have the tmp directory in your configuration.php file(s) setup incorrectly. If that's not it, then there's some sharing between your sites going on that could be causing the reconnections above (one site writes to the wrong database) or a slew of other problems. JFBConnect wouldn't do this without something within Joomla (or JomSocial) being configured incorrectly, and that needs to be looked into.
Hope that helps,
Alex