[debjit]

If you are sure it won't be a institution-level integration this is fine.Otherwise you may probably need a Admin - Teacher - Student hierarchy,but that will probably be years later before we start talking this.

As for the app I'd provide open registration and invited registration.Teachers can signup and create direct invite links for their students to sign up,while new individual can signup normally to create a new class.

Paid features can also include advanced result exporting (more than just username and their scores),simple(in the ease of use) pattern matching for "fill-in blanks"(eg. [answer1|answer2]),more question type like handwriting etc.

And by the way,if a student lost connection while on exam with a time limit,how will you handle it?

*As for the app I'd provide open registration and invited registration. 
Yes, the admin can choose to make registration open or close. There is really a lot of flexibility.

*And by the way,if a student lost connection while on the exam with a time limit, how will you handle it?

This is the only MVP version. People can log in and give exams for the duration of the exam. 

There is not counter cheating mechanism is in place.

This is a bare minimum working Exam application.