Is it true that boys and girls have separate classrooms in Korea?
It's true in some cases. It really depends on each school's policy. There are even some schools that have only male students or female students. This is true for colleges and universities as well. Ewha Womans University, for example, only accepts female students and not male students.
(As a side note, Ewha Womans University was founded by an American named Mary Scranton. She believed that every woman has her own identity and importance. So she did not name the school Ewha Women's University. Instead she used the word "Womans" because presumably she did not like the idea of grouping all women as if they are the same.)