Am I granted the same rights as residence in respect to healthcare.
Healthcare in Poland is based upon the insurance principle, which means that you need to have paid health insurance in order to get treated. There is no universal coverage based on being a citizen alone.
Or on the other hand, if I were to run into trouble with the law in Poland, would I be treated as a Polish resident or American?
Polish, if you've got Polish citizenship, which you have.
I have had such a hard time getting in touch with anyone at the Polish Consulate in NYC.
Should be easy enough to do in person in Poland. Just go to the passport office with your old passport and ask them how to renew it.
The New York consulate is staffed by idiots. Try Chicago.