Bonnie,
Thanks for the access to the site. I meant to get on earlier today, but it had to wait until tonight. However, I think I made some good progress.
Looking through your system plugins, you have a lot. While that's not necessarily bad, it's always best to make sure that you only have enabled what you need. If there are some you don't think you're using anymore, you may want to try disabling them. In general, less enabled will speed up your site and lower the risk of conflicts in the future.
With that said, I initially noticed that you had the JFBC Kunena Profile plugin enabled. I couldn't find Kunena installed though. So, I disabled that. While JFBConnect should properly detect that Kunena isn't installed and simply skip trying to do the profile import into Kunena, it's possible that that was a cause of the 403 failures since the plugin was getting confused.
Next, going back to the "logged in but not really logged in" issue, I noticed that when I clicked the Login With Facebook button, the page reloaded almost instantly. Since Facebook has to be connected, information imported, and other stuff, it simply seemed too fast. So, I went through your system plugins for caching type plugins. I then disabled the "jotcache" plugin, and things seemed to work for me.
Now, Jotcache has some parameters for Browser Caching. I'd recommend simply disabling that setting, for now, if you want to try to enable the other features of the plugin. First though, test with it completely disabled before going back and (possibly) introducing more problems.
Either way, hope this all helps, but if you still run into issues, please let us know!
Thanks,
Alex