Law / Got Polish citizenship, now I need a Pesel and passport [26]

Hi, I'm new to this forum and decided to join for some assistance.
I'll be traveling to the Czech Republic and have plans to go to Warsaw to register my foreign birth certificate and then to get my PESEL number thru the Polish Consulate in Washington for which I'll only need to email or fax my Polish birth certificate so they can apply for my PESEL and then provide me with such number to apply for my Polish passport...

Question is: Do I have to have a registered residence in Poland to apply for my passport there? Being that I'll be traveling to the Czech Republic on a Schengen visa, can I stay in Poland beyond the expiration of my Schengen visa while I wait for my Polish passport? In the process of waiting for my Polish passport, can I apply for my Dowód osobisty?

I have plans to relocate and to stay in Poland but don't want to break any rules by staying in Europe beyond my Schengen visa expiration...

Please some advise is appreciated...close to travel within 20 days.
Thank you in advance.
CK