If your passport wasn't stamped when you entered Schengen, they may pick on you on the way out and want proof of when you entered. Yes they do this, yes in Krakow airport. Just tell them the truth and they probably don't have much of a case against you, plus you're American and all that :) Good luck with it all.
Related: EU Residence card holder - staying in Poland for more than 90 days
I am a Non-EU citizen but holds an EU permanent residence card from The Netherlands. I will be staying in Poland for about 4 months.
My question is, do I need to apply for a residence and if yes, where can i apply?
Thanks for your help!
No you don't need to apply if it's just travel. Your residence card from NL would work for legal travel in all of Schengen as long as it is good for. You cannot work legally in Poland with it however, you need a Polish Work Permit and Residence Card for that, but it doesn't sound like that is an issue.
Here is the contact page for the Immigration Office and Border Guards in Warsaw if you want to verify: