"Puerto Rico isn't part of the United States. It isn't a state or commonwealth but is a country that happens to be under US protection. And suddenly to be a nation, one must have a representative at the UN?"
Umm, Draug it very certainly is a part of the United States. Its a territory. And it certainly isn't a country, because being a "country" implies sovereignty, which Puerto Rico does not have. Puerto Rico cannot make foreign policy, cannot conduct trade with other nations via their own representatives (must go through the US Trade Delegation to whichever country), etc. Also, there are many of those in Puerto Rico who do want independence. They just happen to vote on it rather than fight and lose.
Chechnya is a Russian territory in rebellion. It is part of the Russian Federation, which is a country. Russia is not a country. Its strictly a name thing, like its not China but the PRC, not Taiwan but ROC, etc. Federation is just in the name to describe the type of government.