I don't think you are. I'll take it one step at a time:
1) Good
2) This is normal IF you're logging in through Joomla.. not with your Facebook credentials. The "Reconnect" button is letting the user also login with Facebook, in which case, if they do, they can then later log into the site with either their Joomla account, or their Facebook account. Also, being logged in by Facebook makes it easier to comment or do other things that require the user to be logged into Facebook.
IF you're seeing that message after you've logged into your site with Facebook, that's wrong. I just logged in with JFBConnect/Facebook, and everything seemed right though.
Finally, if you want to disable the Reconnect button, you can turn it off in the JFBCLogin paramters.
3) Are you logged in with Facebook? If you log in with your Joomla account (on this page:
www.inewslocal.com/index.php?option=com_...iew=login&Itemid=155), you won't be logged in with Facebook.. which won't let you comment. Not sure why the login button on the comment box doesn't work. Can you give a page with comments on it for us to test?
4) Check the redirection URLs in both the JFBConnect main component configuration and in the JFBCLogin to make sure they're setup how you want. If that doesn't point out any issues, let us know and we'll figure that problem out too!
Good luck!