You are right at least on one point: no communist country ever experienced immigration problems. Not because they did not want immigration, but because nobody, on his own mind, would have wanted to immigrate to a communist country.
Beside this, if the US keep on letting Mexicans colonizing its territory, the US will become a second Mexico, since a country is worth what the majority of its residents are worth. An other option for the US would be to become a communist country, which would prevent anyone for immigrating to the the US.
Not true. After the division of Germany, many German Leftists in the West migrated to the new East Germany. This was VERY common. Also, I have heard that Cuba has many recent immigrants from Jamaica and Haiti, and they like it very much and say it is better than their own country. Although Venezuela is not a Communist country, it has recently had a large number of poor and working class Colombians moving to it. They say they like it there a lot better than in Colombia. China is currently run by the Communist country, and it gets quite a few immigrants. Currently quite a few Americans are thinking of moving to Cuba if and when they retire. A number of American Leftists have already moved to Cuba and live there currently. Philip Agee is a prominent one. An old friend of mine from 1980 was from the Azerbaijani SSR in the Soviet Union. He told me that Azeri Soviets and Iranians used to go back and forth across each other’s borders all the time.
Most Communist countries were paranoid and they didn’t want a lot of immigration. They thought there might be spies mixed in.