Juraj:
and GPIO0 was grounded for flashing and is pulled HIGH for normal boot?
GPIO0 was grounded during flashing - no effects, as you can see in post above. For normal boot GPIO0 was unplugged, but decided to try high as you mentioned and still it didn't work.
Tried even using random program to check if ESP is visible. Well, it isn't. Beside flashing my module seems to be invisible in serial monitor and other places. The real question, is my device really updated? So it could connect through adapter to get flashed, but can't connect to let me communicate with him via AT commands? Seems weird...