Oddly enough, today everything seams to be working.
Yesterday, while trying to fix it in safe mode, I went to save mode a few more times and tried to find some clue as to what can be wrong. From what I remember:
I checked the log and found nothing unusual, except that I noticed that scheduled scans were running all the time, even on days when I could not access the application window. So it seams that it could be just window / icon display problem, but all necessary services were running in background for all this time.
I played with some other settings, mainly turned on Windows System Try notification on. Usually I disable all pop-up's, but I just wanted to see if it make a difference - yesterday, after forced restart it did not.
I disabled a scan for rootkits. I not sure if it was on or off before, but since there was no effect after changing it I left it in disabled state.
as I was not making any progress, I changed language settings from PL to EN (UK), to make our future communication easy and turn off PC for a night.
Today, when I turn PC on after I was back from work, an icon is in a try, I can click it and it opens application window, generally all seams to be working properly.
I'm kind of skeptical that changing a language could make a difference but at first glance it looks like that.
If it is of any help to you, here are current support tool logs - maybe you can find something useful by comparing them to previous results.
mbst-grab-results.zip