Uwe,
Just got into your site to look around further. What I'm seeing when I try to register or login through Facebook is that first, I'm redirected to your News page, but then a redirect loop starts and I get continuously redirected to /index.php. I'm not seeing the session errors you mention above, but obviously, this isn't good either.
I hit 'stop' in my browser and then looked at the source code and all that was there was your Google Analytics code and a Javascript redirect to your home page, which looks like:
document.location.href='http://blink-182.de/index.php';
That line would explain the continuous redirect that I'm seeing, but I'm not sure where it's coming from.
After I did that, I logged out of Facebook and looked at the code on your home page again (no redirect loop happening this time), and if you look, you'll see the Google Analytics code is being loaded at the very top of the page.. even before the opening html tag. Something is inserting code before the actual web page output, that's not JFBConnect, may be related to your Google Analytics code, and no matter what isn't correct.
Hope this helps get you started, but I think if you investigate this issue further, it might lead you down the right path. Again, I don't think this is related to JFBConnect in anyway, but we'll gladly help you however we can.
Good luck,
Alex