Sure. It actually works using the Joomla authentication and login protocols, so there shouldn't be an issue there, but of course, it is pulling data from elsewhere, which may cause some (easily overcomeable) issues. We haven't tested widely with synchronization plugins, so will need your help (and feedback) through this.
First, ensure that the plugins fire in the correct order.
* For Authentication, if your sync plugin fires, make sure it's after both the Joomla and JFBConnect authentication plugins
* For User, make sure that the ordering is Joomla then JFBConnect, then your sync plugin. Other plugins in between each of these is fine, as long as the chain is maintained.
The user plugin is what provides the Username, Full name, and other information back to the Joomla store in the session. We don't override or manipulate session data directly.
Finally, not sure if you're using "Create New Users" set to yes or no. If yes, the FB user should be treated exactly like a Joomla user. If no, things like password don't exist, since you're authenticating to only an external source (FB). Not sure if your synchronization relies on a password being present, but this could be part of the problem.
Let us know if this explanation helps and/or leads to more questions. Would love to get you going so we can say "Successfully tested with synchronization plugin XXX"
Thanks!