One big step towards a solution was to recalibrate the z-axis sensor. I ended up moving it 0.2mm upwards and the print surface is much better now. I also tweaked the extrusion multiplier by 0.01 because I still have the impression it is extruding not enought. Will dial the extruder current up also. So at the moment I get good prints again. Thanks for all the advice!
@Brook_Drumm What kind of filament do you recomment that is available in Germany/Europe?