Hi Stefan,
I had Problems similar to your’s in the past, my z-axis was moving without g-code to move the z-axis.
This post from @jeffeb3 helped me to find the issue: Z-Axis moving up when in contact with material - #9 by jeffeb3 put a piece of paper on your leadscrew and make a video.
HTH
Cheers from Heidelberg 
and good luck with your problem solving attempts!
felix
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