Why? Every single insurance policy will have a clause regarding the roadworthiness of the vehicle. Mine certainly does - and using summer tyres in diabolical winter conditions certainly means that the car isn't roadworthy.
No. Roadworthiness (i.e, having a valid inspection/MOT) is determined by having a valid badanie techniczny. And you probably know how worthless that is. Nonetheless, there is no law or case law or regulation that one should use winter tires in "winter conditions" in PL From what I remember, there are only two laws in Poland that cover tires: the first is on minimum tread depth and the second on using two tires of similar make/model on one axle. If you meet both requirements, you are "roadworthy". Other normal counties that do not have laws on the mandatory use of winter tires still include a clause that finds drivers at fault for not using them (Switzerland 100%, as I, but even in those countries you could probably go to court and win.
I personally believe that people who don't switch to winter tires and use summer or all-season tires are outright idiots and ought to be shot on sight, but unfortunately that's the way it is.
The winter tyres are working well on snow only.
No. A common and dangerous misconception. Again, winter tires are called winter tires and not snow tires for a damn good reason.