Now I have another problem. Z-axis isn't moving while printing - it keeps printing layers on the same level. In "pronterface" software it works ok, goes up and down when tested, homing works fine. I also tried directly from the printer. Every movement in every direction works but still while printing the Z axis doesn't move.
Maybe it's a stepper driver? Any ideas what could cause the problem?
I noticed you have a lot of mods in place to force the lead screw to not wobble. Lead-screw wobble is a sign that something is misaligned. Mods meant to force the wobble away is like treating the symptom.
In a worse-case scenario, such a solution will put too much strain on the stepper. The motor draws more current to compensate. The extra current burns up the stepper driver.
It's just a guess, but it sounds like this may have been what you experienced.
Much better to get the alignment right and leave the top of the lead screw naked.
I replaced the driver, now it works. But I think it might have been also something in slicer or configuration in marlin. I resetted it once again to make sure it's clean start.
Im sure its not this but have you converted the gcode to a .txt file and checked that the slicer is actually creating a correct gcode with z coordinates?
Check to make sure the grub screws in the coupler connected to the lead screw are nice and tight and aren't slipping.