Thank you very much Samuel Wood (Otto)! Also for explaining everything so good. :-)
It worked like a charm. I already spend too much time on it and even moved servers because I was thinking it was my hosting (just like some others), but I'm glad this major problem is over now. I wasn't sure anymore what to do next, so I'm glad I can put this all behind me.
I just wonder why it takes WP so long to publish a new update with a fix. Or was this just affecting only a few sites?