Secondly, I might be wrong but I don’t think the Norwegian government gives the Nobel prizes, an organization based in Norway does

Correct. The Nobel Peace Prize is awarded by The Norwegian Nobel Commitee, not the government. The committee consists of members appointed by the Norwegian Parliament – again, not the goverment. The commitee thus a private body, and the government has absolutely no say in their selection of the recipient.

That’s just… factually wrong. Greenland is an autonomous part of The Kingdom of Denmark, and has therefore been a member of NATO since it’s founding in 1949.