Housing and Rental affected by Immigrants how much?

Rent has more to do with a community's average income and infrastructure. Cities with high infrastructure and high income are going to be more sought after, and therefore will have less available housing.

If you had a flood of people moving to a low-cost community all at once, that community would likely see some increase in rent. However, that low-income community would also get a boost in its infrastructure due to the increased demand of living there.

For high-income communities, the change is less significant. Immigrants trying to live in NYC or LA will have the same problem everyone else has in trying to live there. The high-paying job will have to come first. If the job doesn't pay enough, then you get a longer commute. Blaming that on immigration seems silly. It's the same old "they're taking our jobs" routine which is usually directed at illegals who you know damn well are not working as advertising execs.

"Typically immigrants work for less which enables higher profits, and they also buy food and clothes from large chains like Walmart so why do we call immigrants “vibrant”? "

How do you define vibrant?

And what does this have to do with high, un-affordable rents in cities?

In terms of 'vibrant' in the economic sense of maintaining the dynamic role of technology in maximizing output, high-skilled immigrants do contribute disproportionately to innovation, seeking patents at a higher rate than natives.

Yes population growth decreases GDP per capita due to constraints on capital volume and falling capital-per-capita as you described ... textbook macro. So are you really concerned about the upward pressure on the Gini? Or are you just concern trolling? As a property owner I'm not seeing your implied downside.

Immigration needn't reduce per capita GDP. It could just as easily be used as a tool to boost per capita GDP should the policy focus on adding greater numbers of skilled immigrants to the labor pool.