I would be really happy for hints what to try next. Here's update:
got a mobo to try - Gigabyte EX58 UD3R and random shutdowns continue -> so it is not bad mobo
got different RAMS - Corsair TR3X6G1600C9 (3x2GB 1600MHz) and random shutdowns continue -> it's not the memories
(I tried different Power Supply and Graphics card before and no change)
The last piece is CPU, can it be that??? (At the store they say they doubt it). My BIOS is F3, and I am scared to change that. Can it be that?
I'm really frustrated, so any help appreciated. Thanks a 1M.