Sorry for saying EasySocial instead of JomSocial. We have similar integration with both though as JomSocial has a "Integrate with JFBConnect" option in it's configuration area as well
If you're still trying to get it setup, let us know. There are a lot of settings in both JFBConnect and JomSocial that could be at play here. For #2, I know the JomSocial Redirect plugin has caused issues in the past for behavior like you describe, but it could be something with SEF functionality or the menu item you have setup for the New User (or Returning User) Redirection in JFBConnect.
For the logging in part, if you can let us know the specific message, that will help. It may be related to #2 though in that the page being redirected to is simply restricted to another level of user so you're getting a message that seems like the user isn't logged in when actually they're trying to view a page they don't have permissions to view (which can look like a page that doesn't exist).
I hope that gives some ideas on possible causes, but like I said, I can gladly help further if you're still trying to get things going!
Thanks,
Alex