Z axis calibration issue due to X axis

I finished the build of my first 3D printer (Prusa i3) a couple of weeks ago. In general the printer seems to be working. However, I am noticing a strange behavior that affects the first layer very much:

I calibrate the Z axis in the middle of the plade using a sheet of paper. Now I can move the Z axis up and down (e.g. +120mm and -120mm) and the nozzle reaches the exact same height again, as expected.
However, when I move the X axis (e.g. homing and back to the center again), the distance between the nozzle and the plate may change between +-0,2mm without the Z axis motors turning. This change in height is obviously too much and often ruins the print due to a bad first layer.

I have tried a lot and don't know where where else to look for the cause of this problem.
What I have noticed is that the LM8UU bearings for the X axis seem to have a little play and cause a vibration or metallic sound at certain speeds. The printer has been doing this since the beginning, I even changed the bearings without any change. But thinking of the extruder's weight, this should not be an issue?

Any help with my problem is appreciated.

Regards,
Bastian

Does the height change depending on the position of the extruder on the X axis, or does moving the extruder in one direction then back to the original position change the Z height? If the former, the bed isn't level. If the latter, see if you have an movement in the extruder by attempting to rotate it by hand along the Y axis. If you get any play, tension in the filament may be pulling back on the nozzle, or release of tension allowing it to rest back down, as you move the extruder along the X axis. Consider changing the position of your roll of filament to reduce tension.

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The Z height changes even at the exact same X axis position. So an uneven bed should not be an issue here. There is some movement visible while rotating the extruder. Also some play between the smooth rods and the bearings is visible. Tension of the filament could be an issue in general. But when moving the extruder back to the same X position, the tension should also be the same. When I reduce the tension to a minimum I still get experience the problem.

Can you post a picture of the printer and the extruder block? I might know what the problem is, but I need a few pictures.

I uploaded some pictures CLICK HERE. Let me know if you need more specific pictures.