how do public schools function within the great kingdom? are they required by the world anatheum, and do they have jurisdiction on how they function, or is that solely up to the great kingdoms?
Good question! This is up to each individual kingdom.
Folkvar: School is not free unless it's military school. Military school is open to all species from childhood to adulthood, so long as you can pass boot camp. Though children can attend military school, they can't actually join the military until they're adults, whatever that means for their species. There is a catch: if you join military school, you are required to serve in the military as an adult for at least 5 years. You can't just join, get a free education, then peace out without giving back to the government or you will be charged with treason. Other types of school are not mandatory in this kingdom, so literacy rates are pretty bad. Folkvarans can barely spell their own names, but they sure can swing a weapon!
Matuzu: Basic schooling is free and mandatory for everyone from toddlerhood to early adulthood. Basic arcane schooling is free for fae and gaians up to a certain age. College is not free, but the government has scholarships to reduce the cost for many people. Matuzu Kingdom invests a lot into its education system and it shows, at least in urban areas. Some of the rural regions can be sketchier in this regard.
Lamai: Basic schooling is not free, nor mandatory for anyone. But you know what is free? Serpentist religious school! Religious school is not mandatory, but because it is free, many Lamaish citizens end up sending their kids there anyway in hopes that they'll learn something useful. Kids pick up basic reading and writing skills here at least, which they need for their religious studies. This religious school is free from toddlerhood to adolescence.
Yerim-Mor: Yerim-Mor once had a robust education system that rivaled Matuzu's, but since the war this system has completely crumbled. The government has cut funding from education entirely to address more urgent things like food and water. So, any schools here are privately operated, meaning they usually cost money and have their own set of rules. Some of these schools are run by the evil Cult of the Crescent, and there is nothing the government can do about it.
Zareen: Schooling is free and mandatory for everyone, but only from childhood to adolescence. The exact age depends on species, since lifespans can vary. Humans, for example, are free to drop out of school and join the workforce at 13 if they choose. Satyrs and trolls may do this at age 11, and elves at 16. There are mixed-species schools as well as species-specific schools, but they all operate more or less the same way. College is not free for anyone and scholarships are hard to get.
Evangeline: Basic schooling is free and mandatory for all human male citizens age 5-15. Other commoner species may have to pay for schooling, depending on what's available in their district. There are human-only schools, inhuman-only schools, and mixed schools in this kingdom. But you will never find fae or gaian schools because it's illegal for those species to attend school at all.
Mogdir: Basic schooling is free for all species from childhood to adolescence, but it is not mandatory. Basic arcane schooling is also free up to a certain age. If a student wants to further their education in either mundane or arcane studies, they have to start coughing up money for it.
Etios: School is not free here, but it is not mandatory either. Many Etiosi live their whole lives without attending a single class. Schooling is paid for on a year-by-year basis, so a citizen's level of education usually correlates with wealth. Schools are segregated for gaians and commoners, while fae are not even allowed in this nation's borders.
Seelie: Basic schooling is free and mandatory for all fae from childhood to adolescence. Arcane schooling, on the other hand, is not mandatory or free. Seelie schools are segregated for fae and monsters. The Seelie government oversees fae schools, but it does not oversee monster schools. If monsters want schools in their communities, they must fund and run them themselves. Drau and adhene have done this pretty successfully, but pixie and kobold communities are still severely lacking in education. Commoners and gaians are considered "wild animals" here and may not attend any kind of school.
Unseelie: Attending school is not mandatory here for anyone, and it's usually not free. That's because schools here are run by private entities and are not overseen by the government. This means that schools can charge whatever price they want for attendance, exclude anyone they want, and make all their own rules--for better or worse.
Damijana: Basic schooling is not free, but it is mandatory for all red elf citizens age 8-30. It is also mandatory for other species from toddler age to adult age. Mandatory but not free? You heard right, you can actually go to prison if you're too poor to pay your kid's school fees every year! Damijana just loves slaves--I mean, prisoners...Most schools are for elves only, and many more are specifically for red elves only. Non-elves must attend mixed schools, which are always terribly underfunded and attended by the poorest citizens. Red elf citizens are entitled to a few of free college courses if they graduate basic school, though they can only choose from police/military, business, or nursing. Anything beyond that is not free. Other species must not only pay for college, but they pay an inflated price compared to what native red elves pay. As you can see, it's very difficult for non-elves to get a decent education in Damijana.
Alliance: Basic schooling is free and mandatory for everyone. Arcane schooling is also free and mandatory, but only for Aquarian species. Non-Aquarians have to pay for it. Some mundane and arcane college courses are also free for Aquarians, though cecaelia have more options available to them than anyone. Cecaelia are eligible for the most scholarships, and the system is stacked in their favor overall.