If a 'rich fucker' lives in a house that is valued at twice the average value in their community, then they pay twice the property taxes to help fund public schools their 'average' neighbor pays. (There are no deductions or loopholes.) If that 'rich fucker' then turns around and enrolls their child in a private school they are 100% responsible for the tuition costs and get NO deduction or credit on their property taxes.
That depends ENTIRELY on the state. There are plenty of states that give the parents vouchers to send their kids to that private school. So no, they aren't 100% responsible for the tuition costs, and they do get a "deduction" or whatever you want to label it, in the form of a voucher.
The real motivation for change/improvement in public education will be school choice/vouchers - that will allow concerned parents to abandon failing public schools for better ones, and as failing schools are shuttered bad teachers can be weeded out of the system. Competition is healthy, the lack of serious competition is (contributing to) our currently failing public education system.
No, shitty parents are why our schools are failing. There's absolutely nothing a school can do to cope with parents who don't discipline their children, or make any attempt to help them with their homework. There are only so many hours in the day, and at some point a parent needs to be a parent. When all those "failing public schools" close down, and the state mandates private schools not turn children away, the same shitty parents with the same shitty kids will cause the same shitty problems in the private schools. You're delusional if you think otherwise. Unless of course you're advocating that private schools should be allowed to deny students access to an education after all of the public schools have shutdown. In which case you and your elitest worldview can pound fucking sand. When the poor revolt don't expect anyone to rise up and defend you.