(I assume you follow FIRE, and know that this problem isn't contained to the "speech is violence" people; there are numerous right-wing attempts at suppression on campus as well.)
As to your second point, FIRE tends to advocate for free speech on campus generally, including private schools. A lot of the right-wing attempts at suppressing speech happen at places like private religious schools (Duquensque and Georgetown are two top posts in the FIRE website). I tend to stop short of that—I think private schools are free to restrict speech in ways public schools are not. But, as I said in my post, attempts to suppress speech have historically been a right wing phenomenon.