I couldn't remove the backlash. I tried several things, but none of them worked. For whatever reason, the resistance to overcome the backlash of the ballscrew is very high.
The print artefacts mentioned in the last log came from the Z-motor, which lost some steps. So I adjusted the motor current. Now this is fixed.
I printed some 3DBenchys, and the print quality looks very good.
From some light angels, you can see layer "misalignment". It looks similar to the "error 602" from the Prusa i3.
Perhaps it is due to the small difference in filament diameter (+/-0.05 deviation). Or can a mechanical problem be the reason for that?
The biggest problem I'm facing right now is the inaccuracy of the Z-homing.
When I start several prints one after the other, the distance to the bed for the first layer gets smaller with every print. I don't know what the reason for that is.
The only things I can imagine are:
The axial-bearings are frictions ones, so maybe they need some kind of "warming-up" (some movement), so they are working more freely and do not get "stuck" in motion.
The second reason I can imagine could be the mounting of the endstop. Right now it is mounted at the bearing-housing of the Ballscrew which is mounted to the frame. The problem is that this bearing-housing moves slightly while the printer is in the folded position and tries to unfold. This makes the homing-process probably more inaccurate.
The housing moves due to "extreme" forces while the unfolding-process. The forces cause the Z-motor mount to bend slightly.
Through the bending, the whole assembly (Z-motor, ball screw and bearing-housing) moves a bit towards the front, because these components are all connected the bearing housing moves as well. It would be better if the endstop is mounted truly fixed to the frame(I'll try that next).
These are my ideas about this problem, but I'm not an engineer. I'm open to helpful comments about that.
I uploaded the whole assembly of the X printer(CAD-files) on the Github repository.