Steffen,
Apologies for the delay. We tried to recreate the problem on our development servers (to avoid too many changes on your site), but were unable too. Then, I was travelling for 2 days, which limited my support abilities.
I just got into your site to test some things, and it's very strange what's happening. Basically, if we set the Content plugin to show the like button at the top of the article, it inserts it into the top of the article, but also right before the <html> tag at the top of the page. If we set it to the bottom of an article, it sets it correctly at the bottom of an article as well as after the closing </html> tag at the very bottom of the page.
We've never seen this before, and again, can't recreate it. It's like there's another plugin installed that's firing our content plugin and using the text of the whole page in addition to it firing normally and just sending us the article contents. Does this make any sense to you, or help hint at something you may have configured to do something out of the ordinary with articles?
If you don't, then the next steps would be to actually do full debugging on your site. If you have a development site to work with, that's obviously best. For a few minutes, we may need to 'break' the page to print out specific statements and determine what's going wrong. Let us know if this is ok, or where the best to do this debugging would be.
Sorry for the troubles. We'll get you fixed up soon.