Important update. I think my Intel NUC DC3217IYE was BROKEN from the beginning. I came to this conclusion when I noticed that the YouTube Flash player also crashes without reason. It was confusing as the HTML5 version of the YouTube player does not crash and there are no other problems apart from the RME Babyface problem. So I tried my Babyface with another NUC DC3217IYE and it works there.
I'm sending my broken NUC for repairs.
EDIT: After all, I found out the problem was caused by a compatibility problem between TP-Link TL-WN722N USB WiFi Adapter and Intel NUC. Almost impossible to see the connection between video card and audio problem relating to WiFi adapter! Especially as I needed that internet connection to test YouTube. But the culprit was TP-Link (what an amount of wasted time because of that).