Speaking of Polish Anthem singing, Are you serious about it?
I don't think you need to sing the national anthem to get the residence card. I have a residence card but never had to sing it.
Here's the link in English about the residence card. You can click to download the files from "Text Attachments" at the lower part of the page.
Firstly you have to get a "Residence permit for a defined period of time", and when this card expires(in 3 years I believe) you're eligible to obtain "Permission to settle".