/ How long can you drive a car registered in another EU country in Poland?
If you are in Poland and you bought the car on foreign plates, you have 30 days to register car in Poland. If you are from EU and you've got a car registered in EU I don't see any problem with driving in Poland ;) Just remember about your driving license, ID and registration document (of your car) together with liability insurance. Some people register new cars in Czech Rep. or Lithuania to pay less and they can drive without any problems.
About police hmmm, if you don't brake the traffic law, no one will care about you (unless you've got German plates, some Poles buys cars form Germany and they do not register them in Poland). You can meet many different foreign plates in Poland, e.g.: Swedish, Norwegian, Russian, Belarussian, Ukrainan, Lithuanian, Latvian, Estonian, Slovak, Czech, Spanish, British, German etc. Fines are the same for everyone, do not try to bribe police officer (you pay fines via bank transfer, if you are not sure how to do this, you can go to post office, they take about 10pln for that) unless you know someone.
If you want to register a car, check this: europa.eu/youreurope/citizens/vehicles/registration/formalities/poland/index_en.htm