Hi Sarah ~ !
1) According to the referenced link, the book marks down the names at the first proper sign of magic, which could be as late as six or seven, so I suppose it's reasonable to say that you wouldn't necessarily have to have been born in a country to attend the corresponding school. You would, however, need to have rather permanent residency from an early age I reckon in order to be consistent. I would imagine that once someone's name is recorded at
any magical school, it sort of "goes off the record" to avoid having people claimed at multiple magical institutions. It's also worth noting, however, that said book resides at
Hogwarts, and it's possible that the system for admissions at other wizarding schools is different. Generally speaking, I would say as long as you have a well-considered and thoroughly explained reason for having your character attend a certain school, then it's usually possible to work something out. It's hard to give a concrete yes or no without all the details.
2) As per this
announcement, if you desire to develop a non-played wizarding school, please contact Fe
e with your ideas. I imagine home schooling is possible, though depending on the country would probably require special licensing from the appropriate Ministry of Magic. If, for example, your character was nomadic in lifestyle and didn't really belong to organized society, that would be different than say someone who lives in a regular city but is otherwise not economically well off. As far as
paying for school, canon says this about Hogwarts
"There is a fund at Hogwarts for those who require assistance to buy books and robes. You might have to buy some of your spellbooks and so on second-hand, but –"
Half-Blood Prince - page 256 - Bloomsbury - chapter 13, The Secret Riddle
so I imagine the same applies to other magical institutions. Similarly, since magic runs in families, it's highly possible for students to use second-hand (or hand-me-down) equipment at schools. Not everything would need to be new.
I hope that helps!