If Twitter is all that shows there, it means that your account isn't connected to a Facebook account. I'm not sure why that would be, but the only way that connections are deleted are:
* If an admin has deleted the connection manually from the user map area
* The user has been deleted from the User Manager
* The user, on the front-end, has re-associated their Joomla account with a different Facebook account. In this case, the connection between FB user and Joomla user will be changed, but it should still exist in the admin area.
Again, there's absolutely no link between a user's password, either on Facebook or your Joomla site. The link is completely based off the user's Joomla ID (which should never change) and their Facebook User ID (which never changes).
From what you describe above, it seems there's no connection between the Joomla user and Facebook and you'll need to make that connection.
Of course, if you make the connection please test anything you think may be causing problems. If, after a successful connection (you can login with that user in FB and the connection shows up in the User Map area), try changing any passwords and make sure things are still linked. We've never heard of any issues like you mention, but if you come up with a way that causes issues, we'll gladly investigate further.
Thanks,
Alex