[MysticCat]

^^^ If by "lower level" you mean public elementary, middle and high schools, then yes and no as to having the same rights. The courts have recognized basically an in loco parentis (in the place of the parent) theory as to some rights, so that, for example there is a much lower expectation of privacy (lockers and bookbags can be searched more freely than would be the case in the "real world") and dress codes forbidding, say, certain things on tee shirts can be enforced.

But on the other hand, the First Amendment gives rise to the whole line of cases about teacher-led prayer in schools and prayers at events like football games and graduation ceremonies. And it's been clear for a long time that, due to the First Amendment, students cannot be required to recite the Pledge of Allegience (or ostracized for not doing so).

I don't think there's been any lack of parents or students going to court regarding these rights.