I just tried multiple things to understand what was wrong with Facebook, but wasn't able to find the cause. The only things I've determined is that our Javascript is not fully executing properly. There's a couple causes I can guess for this, but I don't know the exact solution since I don't know all the extensions on your site and how they may be interacting.
What I can see is that:
* When the page loads, there are 2 Javascript errors:
Uncaught TypeError: $(...).dropdown is not a function
www.tricolordogs.de/:578
Uncaught TypeError: Cannot read property 'msie' of undefined jquery.fancybox-1.3.4.js:29
I'm not sure what's throwing either of those errors.
* jQuery is included 3 times on the page. These are the different inclusions:
<script src="/media/jui/js/jquery-migrate.min.js" type="text/javascript"></script>
<script src="http://code.jquery.com/jquery-latest.min.js" type="text/javascript"></script>
<script src="/components/com_imageshow/assets/js/jquery.min.js" type="text/javascript"></script>
Each of those inclusions is in different parts of the page, which could be causing conflicts to other Javascript features, like JFBConnect.
I can't guarantee those are the issues, but fixing the Javascript errors and reducing the jQuery inclusions will improve page speeds and help narrow down what the actual authentication issue is.
I hope that helps you, but if you need more help, please let us know.
Thanks,
Alex