To do a banner on your own should be pretty simple. I'm not sure how you're creating that banner now, but hopefully it doesn't have to be through the same method..
What you want to do is create a new "Custom HTML" module. Place it in the Banner 3 (or Banner 2, it looks like) position. When editing that module, there's a big box for free-form text. From here, you can do what you want as far as text goes, and wherever you want the Login With Facebook button, simply type { JFBCLogin}.. You can insert images and do anything here that a website can normally do, but just make sure you don't go beyond the dimensions of wherever you're putting it:
No you can log into CSF with Facebook
{ JFBCLogin}
Hope that helps get you started. Something else you may want to look into is Advanced Module Manager, which we highly recommend. It's very easy to use, and will actually let you specify that the new module you created should only be shown to guests and not users who are logged in. For more info:
www.nonumber.nl/extensions/advancedmodulemanager
As for requiring the user to be logged in, that's definitely not right. I couldn't see a login button on the site to test with, but generally, something like that is caused when the JFBConnect Authentication plugin isn't enabled or the JFBConnect User plugin isn't enabled or is ordered improperly. Check to make sure that it has a higher number than the Joomla User plugin in the ordering column. If that fix it, let us know where to log in (or get the module above working), and we'll see what we can do.