Do I have to present anything to justify that its my gf's car?
no you just have to carry 'dowód rejestracyjny' registration document and the insurance policy confirmation (even in someone steals a car there is a very low chance they will get hold of the 'dowód rejestracyjny' because drivers usually carry it with themselves and not leave it in the vehicle)
the girlfriend doesn't need to be in the car with you, they will only check with your girlfriend if they suspect the car is stolen which is not very probable if you have the 'dowód rejestracyjny' with you)
I don't know about Straź Miejska to be honest - but I suspect their offices don't work like the police i.e. 24/7 - what do you need straż miejska for?