DY you wrote “Can a school turn down the child of a family which won’t keep their television/internet/tznius/shmiras Shabbos/kashrus standards? Can they nevertheless accept a donation from someone who doesn’t keep those standards?”
As a private institution it CAN turn down the child. It CAN set the standards it feels necessary. Neither JUSTIFIES it taking not-frum-enough money from those it deems below its standards of holiness.