How they become students is kinda PLOTWISE up to you. Some people hitch-hike their way to the institute and just turn up, some people are brought in by xavier -- we also threw up sanctuary for this purpose so say if something traumatic and mutant related happened when someone's powers manifested, you can say sanctuary SWOOPED IN and cleared up the mess and took them to the institute.

AND yes pretty much. Unless you... do not want them to idk somehow