I think you'll need to turn that PWM into an actual DC voltage before feeding it into the transistor.
Couldn't you explain then why PWM is not sutable here and what will change after converting the signal?
The transistor needs to be fully on or fully off, not somewhere in between
Wait, what do you mean..? I've seen several project where people use transistors to smoothly control the speed of motor/light of a LED.
The base resistor value seems too big for the gain of the transistor to me
Actually, I first connected an 1K1 resistor, but could hardly turn the transistor on. As this is the smallest resistor I have, I connected the transistor directly to pin (the maximum base-emitter then current makes up 60mA)
When you say the transistor is fully open do you actually mean fully closed or fully on?
I mean fully on, sorry 
Thanks a lot for your replies, but, however, I still have no idea why a load on a motor makes the transistor fuming 
(with only 500mA collector-emitter current). While without any load and the same current it doesn't even become warm.
The problem is that everything works just as expected. Right until I put a wheel on a motor =.