@korinian... I'm afraid I'm stumped.
FantasticFiction is the website I use all the time. I've assumed you used the image at this address:
I've just tried the copy/paste process on a test book of mine and it worked fine. I can't get it to NOT work. My browser is Firefox and I'm using Windows XP. I wonder if this is relevant?
I would think the mobi/specific book you're using is not an issue. If the cover.jpg is not saving correctly into your library directory then it won't work in a conversion either.
Have you tried a different Calibre book and a different webpage image? Does that work?
I've just tried drag-n-drop from FantasticFiction to Edit Metadata. This also works for me.
I've also repeated the same actions after bringing up Fantastic Fiction in Internet Explorer 8 rather than Firefox. Both methods worked correctly again.
Let's hope someone more technical can chip in with an explanation.