The analogy of the house does not fit very well (countries do not invite foreigners, they simply accept or reject their entry), it seems to me better to take it as a "party", at a party you are going to accept well-behaved guests and you are going to to turn away guests who misbehave.
The guest will pay the same taxes as the citizens, so the house is also paid by him.
Again, I did not find any arguments from the philosophy behind the current migration system
This isn't good reasoning. Each immigrant still has own motives and his own thoughts. A large number doesn't render these individual immigrants as a part of collective formless blob or a de facto invading army.
Consider that not all immigrants leave their home countries because of their governments. Many leave because of religious/cultural persecution from one's local community (e.g. homosexuals) or unacknowledged rights (e.g. abortion) in an otherwise secular and ostensibly stable country.
I think you're misreading Nozick. Nozick would suggest open borders, extenuating circumstances notwithstanding. The argument for closed borders would imply that any authority over a country's or state's territorial holdings would belong to the nation's government rather than its people.
> The US and Western Europe are violating the rights of their own citizens by allowing mass immigration.
I don't think GP was talking about Europeans invading America. Instead I think he was talking about two examples of immigration crises:
1. Immigrants crossing the US's southern border arriving in the United States
2. Immigrants from Eastern Europe arriving into Western Europe.
I believe GP is implying that immigrants don't pay their fair share for the house.