Yes. I went to Catholic elementary school, then to a Jesuit high school, and went to a Jesuit college for one year (and then transferred to another university for reasons having nothing to do with religion). And it didn’t even cause me to lapse.
Of course not.
Sure. And I absolutely support their right to send their children to religious schools. Including your hypothetical Islamic school.
But when we’re talking about conservatives (as opposed to religious people who want to educate their children in their faith), it seems that conservatives don’t just want to send their kids elsewhere, they want to abolish public schools. And, of course, teachers’ unions.
Absolutely true, of course.