Actually, they pool their money and purchase homes even more expensive. You'd be surprised. And even if they don't,they support high prices by paying rent to investors, who get a rate of return on those houses in those less than desireable neighborhoods, which supports prices in nicer neighborhoods that more affluent folks want to live in.
Of course not. But for how long? My grandparents [not illegal but vile Mediterranean immigrants nonetheless] settled in the older run down sections of town. Now me upwards of 25 of my cousins live in $200k to $700k homes. We count one millionaire amongst us, too.
Take my subdivision and the surrounding area (a master planned community) as an example. Per capita income within the master planned community is 3X the national average. In my subdivision, the average home size is 4000 square feet. Each house also has garages that hold 2 to 4 cars. Most houses have pools. My next door neighbors on one side are from the Middle East (Christian). My neighbors on the other side are from the Philippines. The people directly in back of my house are from Taiwan. Sprinkled throughout my subdivision are people from other places -- China, India, you name it. None of these people were born in the US, although I would suspect by now that many are naturalized citizens. And none of these residences in my own subdivision, as far as I know, holds more than one family. I estimate that 20% of the homeowners in my subdivision were born in some other country. And it is the same throughout the other subdivisions of the master planned community.
A lot of the people that come here from elsewhere work hard and they work smart. They work their way up, and they take hold of the American dream. The US is the recipient of a lot of the brain drain from other countries, and the trickle-down effects on our economy of their consumption, as well as the taxes they pay, is positive.