I have posted my problem on the AskUbuntu forum, but I would also like to post it here, because there may be people with more experience here! I could also copy the whole question in this topic, let me know if that is desired.
nvidia-bug-report.sh as root and attach the resulting file here.
Just when I was about to attach the bug report, I got it to work! I am stupid for only doing this so late, but my monitor was connected to the hdmi port of the motherboard, and not of the GPU. Just moving the cable to the GPU port fixed it, because the drivers were working correctly… I hope no-one else has to go through this!