Unlimited Drive storage is offered as a part of Google Apps for Education. This is not something an individual student can obtain; rather, school officials enters an agreement with Google on behalf of the school. After that, all students have access to unlimited Drive, etc. All that a student can do is to encourage their school officials to do this.
So, the answers to your questions are:
- No, students do not get unlimited storage if acting as individuals.
- No, this is a form for a school official who will administer Google Apps for their school.
- Not applicable, since students do not sign up by themselves. There is probably some substantial time between filling out the form in 2 and domain creation (since someone from Google will need to contact the school officials and create a formal agreement). After the Apps domain is created, students get unlimited storage as soon as they create accounts on that domain.