Update: It seems that my problem is solved, but I can't say why exactly.
I did a quick PC format, installed DU, tried it and it was still crashing. But then I decided: f**k it and did a second format, but a little bit more carefully. Started with cleaning my PC and reconnecting every part again. Then I installed every possible driver by hand (NOT automatically by WIndows 10), updated BIOS etc. And now it seems to work! I relogged many times, played a bit and not a single crash or disconnect. In the meantime, I switched to a new 1440p monitor, but I don't think it is a factor in here. Anyway, whatever helped, it made a huge difference, because now the game works waaay smoother (despite increased resolution!).