I'm not exactly sure what your asking, but I'll definitely agree the Facebook API can be confusing. Fortunately, it used to be much worse, so they're making improvements!
As for your question, are you asking if they can log into your Fanpage and then be automatically logged into your Joomla site? Unfortunately, no.
You should be able to add plenty of links to your site from your Fanpage. From there, they can log into your site with their Facebook credentials easily, and you can use that information in any of the currently (soon to be expanded) Joomla profile components we support.
If you could explain more your purpose, we might be able to help more. Basically, if you want the user to log into the Fan page, and then be logged into your site, how do they get from your Fan Page to your site? Do you want a "Login Using Facebook" button on your Fan page, that, when clicked, will redirect them to their account on your site?
I don't think that's possible, but once a user does register on your site with their Facebook credentials, then JFBConnect does have a setting to automatically log those users back in when they return. In that case, after the initial registration, if you had a "Go to our site" button your Fan Page, when a user clicks it, they'd go to your site and be auto-logged in. For users that haven't registered yet, you can redirect them to a "Register Now" page with information on how to create their account (basically, click the "Login Using Facebook" button on your Joomla site).
Hope this is sorta clear and helps out. Let us know your specific goals, and we'll try to help get you as close to them as we can!