The button is missing from that page, so I can't see what you see.. as for other notes:
3a) Sorry, yes, I mis-spoke a little. If a user ever goes to a page that is marked as registered (either the menu item is or parts of the component force the user to be logged in), Joomla will redirect the user to the login page. After they successfully login, Joomla (or JFBConnect) will redirect them to where they were before. You're correct that the menu item will be hidden from users that aren't logged in when marked as registered. However, if you then link to that direct menu item location in a button or even another menu item (creating an alias or External URL link), Joomla will kick in and force the login.
3b) Not sure why you're getting the redirect problems. Do you see that issue on your home page when using the SCLogin module? I also have no idea why the invalid token error would present itself. If you're using caching on the site, try disabling it, as that may be interfering.
Finally, another thing I noticed is that it seems like you may have other Facebook extensions or integration going on on your site that could be interfering. The page below has a Login With Facebook button that isn't from JFBConnect:
readysettest.yousaygo.com/index.php?opti...yuc&controller=event
Please try to disable other integrations. If they are loading the same libraries as us or trying to do the same things, bad conflicts can occur.
Hope that explains a bit more,
Alex