here I am asking questions no one probably will ever ask.
1 - Xavier has inherited wealth and he's used his powers to figure out which stocks to invest in and he's got several corporations, all of which goes to fund the institute and the X-men. His current worth is roughly calculated to be $3.5 billion.
2 - Again, rich white guy with ties to government and military. Plus many officials in the government have probably been helped by the X-men or the X-men have saved the world enough times that they're willing to overlook it. Also, it's a private institution for gifted people. There isn't a blinking neon sign over the school saying "MUTIES HERE!"
3 - He has. Several of the more senior members of the team often live off campus. But the school's defences are usually strong enough to deter all but the worst attacks.
4 - Waivers. But more realistically, a lot of mutants who end up at the institute have often been rejected by their parents or have had to run away from home due to their powers. In some stances, their parents are dead (possibly because of their powers. E.g. that one mutant kid in the Ultimate universe who vaporised everyone in a several hundred foot radius when his powers activated...meaning he killed hundreds, if not thousands of people at once).
5 - Generally speaking, mutant powers manifest when someone either undergoes an extremely traumatic event or when they hit puberty. Some people are born with their mutations, but they're currently the minority. So roughly speaking, the students at the institute (in the 616 universe, I can't speak for the TNH universe), would be 12 to 18.
6 - Given the TNH's nature, the institute is a third level education facility, so they'd probably be college age. But it wouldn't be unreasonable for Xavier to have a high school facility separate from the university.
7 - Due to the size of the school, I would imagine that the school has a generalised curriculum, with specialised subjects being done at schools the Xavier Institute has affiliations with, like potentially Empire State University offering science courses.
8 - Wolverine's a history teacher. Seriously, several members of the X-men are actually accredited teachers.
9 - Substitute teachers exist and I also imagine the X-men would have course work prepared for if and when they need to save the world.
10 - In the comics, there was a human nurse. So yes, there can be non mutant staff.
11 - I would imagine courses are year round and I mean year round.
12 - Only a handful of students actually live at the school, but we have some sources citing that about 100 students are enrolled at the school.
13 - Again, only a handful of students actually live at the school, usually the ones too far to commute or are the students at high risk. So they can afford to have one student per room.
14 - It kinda depends. You could potentially go to the school and ask to be let in, but generally speaking the X-men are looking for mutants who are at risk, and they're the ones who are most likely to get invited to the institute because they're the ones who need it the most.
15 - Depends, I would imagine that Xavier's fortune means that he can give the mutants a free education for the ones most at risk. If there is a fee, it would probably close to a standard private school fee.
16 - Considering that many mutants come from overseas, I would imagine they integrate quite well.
17 - No. You'll just be left brain-damaged enough that you'll be able to breathe, but you'll likely shit yourself at least once every day.
18 - Because you weren't lobotomised as a child.
19 - I'm not qualified to make that diagnostic assertion.
20 - See 18.