Thanks for your reply and the tests you already did! I think the Original Files are shot with a Red Camera, some are 6K, some around 5K, 24fps. I don't know a lot because I didn't set up the Library, it was given to me to Edit already prepared. And am also very new to FCPX (one week, so really new). I did phrase my question like that because I found a Question of someone with the same issue on this page, but it was resolved when the Disc with the Original media was mounted. It was some time ago, so I couldn't add my problem to the same Question.
When I send my Project to Compressor I can hear the sound (which is original media) but instead of the picture I see the Text "File Missing". Because of these two reasons I was under the impression that Compressor is looking for the original media and I am missing a setting or something.
Did you also unmount the original Media when you did your tests?
PS: Normal Export directly from Final Cut works with no issue.