I looked at the Google Classroom course document. https://developers.google.com/classroom/reference/rest/v1/courses . It looks the whole google classroom is quite flexible. For example, there is no strict different for student and teacher. It looks there is no way to check the course belongs to which school as well, yes? If admin setup anyone(any google user, not specific to be domain user only) could enrol or become the teacher.
There's no strict difference between teacher and student because you can be both.
Students and teachers are specific mappings between a user profile and a course, representing that user's role in the course. Designations of student and teacher are not global: a user can be assigned as a teacher for one course and a student in another. The designation "student" or "teacher" represents a set of permissions for a particular user in a particular course.
And yes, there's no school name property.