Situation update: So I've spent the whole day, trying to fix it and I run through many possibilites. However maybe I found where is the problem. When I completely removed AMD Adrenalin, I was able to run the games that I couldn't play before. I also noticed, that in windows settings, my monitor is set to 59Hz and when I changed it to 60Hz, black screen and "No-signal" message appeared - same like when starting games.
So this makes me think this: AMD Andrenalin somehow forces monitor to use 59Hz and when game tries to use 60Hz, PC breaks down.
I couldn't find a way to fix it so far. I think there might be some setting in AMD Adrenalin I haven't noticed.