This is hard to believe, but I managed to print something without any issues, for the first time in weeks. And not only once, I did two of these first layer calibration prints in a row, plus a spacer.
Before that, I was cleaning the nozzle on the outside and figured, why not heat it way up, to see if that can get any potential obstructions out of there. After that and a filament purge, I've only seen good prints so far.
Could it be that all of these mystery problems were caused by partial clogging of the nozzle or hot end?