Now I have tried to print again!
I cut off (and filed the end smooth) another 2mm of the screw that goes towards the bottom of the Z position. When I measure from the glass panel up to nozzle 2, the measurement is 305 mm. I tried printing a 0.8mm/diameter cylinder, 300mm high and it seems to work fine now! I hear no more rattling from the stepper motor, which seemed to be losing steps. But I must first, to be sure, maybe print a "real", more complex print before I can be sure. But it might be sometime next week.
Thanks once again for all the help I got from especially @Torgeir
See the video of this morning's print test