Actually, there's a lot of misconception there.
Yes, and you've provided examples, like the following:
We are a country built on immigrants who arrived without English and with different cultural values.
We also had implemented policies to regulate and control who got in, supported assimilation goals, and at various points banned immigration for years as well. We also deported many. It isn't 1720 any more.
You mean like the Chinese Exclusion act and other xenophobic measures? I don't consider that to be a particularly proud moment in our history. That doesn't mean however I believe in utterly unregulated immigration either.
Doesn't matter whether you personally feel proud about it. We also heavily restricted immigration from 1920 to 1965, which allowed the assimilation of the millions who poured in from the 1880's to 1910. Yes, it takes that long for assimilation to take hold. We are under zero moral obligation to take anybody in, nor allow any immigrant to remain, either. We take in far, far more than our 'fair share'; the pro-refugee bleeding hearts need to start sniveling about all those other countries out there to start taking them in. there are plenty of Muslim countries Muslim 'refugees' can go. They aren't' our problem.
I disagree. We have a moral obligation to take in those less fortunate, who wish to come, work hard, and make a new life for themselves. Not in unlimited numbers, but the moral obligation is there. That applies to any religion. We do not, fortunately, exclude immigrants or refugees on the basis of religion and I hope that never comes about. People who's ancestors found a new life here are quick to slam the door.
It's arguable as to whether it is our problem or not....our actions in Iraq began the destabilization that led to the rise of ISIS. We bear some ethical responsibility.