No. The students are limited to the items listed in the rules. If a student is carrying a smart phone, the phone should stay out of sight, and avoid a disqualification for the team.
The height will be measured to the top of the tennis ball.
Yes. If the build materials are provided by the supervisor in a bag, box, or anything else, the container is part of the build materials.
They could get any number of the items.
Remember that the supply kit is designed to create roughly 10 practice sessions. That should help determine how many pieces they would get during a build. Even if you haven't purchased a kit, the list of materials and quatities that is detailed in the Quick Start Kit ordering system might give you some idea.
Also, this is a "mystery" event, so the kit supplies are not specifically what your student might receive.
The students will build their towers on the floor.