this is what the author answered in a FAQ:
The author enjoys coming up with the settings, even if he can't make them cohesive. The boys still have classes, and the fights are basically their extracurricular activity/ afterschool hobby. The classes are just not the main focus of the story. The anime seems to address this topic, and maybe it will expand on it, but the manga is about the delinquent life of the characters, not the classes. Regarding adults being 'absent' or letting kids handle their issues, its shonen manga, a huge chunk of it works the same way.
Furin has regular classes, after-school club activities, and tests. The author says it's because Ume would emphasize, "Make sure those aspects are done properly too!"
But keeping the streets safe and getting into fights are like after-school activities for the Boufurin members. Author also says this kind of school life was only instated after Ume became the leader in his second year
I don't know if I already asked this, but where are the teachers?