Be careful around the central market as many men approached me asking if i wanted to exchange Dollars or Euros. Better not to run the risk of getting ripped off and use a valid exchange booth or better a bank.
Before going to this trip we've heard the rumours that the police in Kaliningrad region is tough for foreigners with giving much fines actually for nothing. And we've survived such an accident just after we crossed the border.
We had all the needed documents, were fastened with seat-belts (in front of the car) but we needed to fasten our passenger seating in the back of the car even if the car hasn't invented the seat-belt in the back of the car! Our fine was 500 Rub, but our good friends at home told us always to have 10 $ in a pocket (it would be about 270 Rub) for such accidents and we were free. Yeah, they earn from the foreigners (possibly not every policeman, I believe there are the good ones too), but if you want to continue yout trip without problems always have 10$ in a pocket and just smile :))
If the driver is a girl, make a flirt, if you are a guy, talk about the football, maybe that will help.