There is no way if the voters vote down a tax increase is the state going to pay, other districts will do the same thing, they will all vote down their school budgets and make “others” pay.
The suing is more complicated and I cant really comment on a paragraph or 2 since I am more familiar with how it works than most. I can tell you that its much harder in general to sue the local board of Ed in most suburban Districts than it is in East Ramapo.Try suing in the 5 towns or Teaneck, you are unlikely to win what you get in EP (Or for that matter NYC, which is a bit easier to win) Most Suburban districts would not give you want people want in ER