What is the status of privacy and freedom from unreasonable searches and seizures on a school campus?
I concur with the above answer, and I would emphasize that the courts are quite generous in their interpretation of what would be considered "reasonable" in terms of a search, especially if it is a search of a student's property - backpack, notebook, car. They are somewhat more restrictive on school personnel searching a person, and for this reason, most schools have made it a policy simply to isolate a suspect student until law enforcement arrives to carry out a search if they see fit.
As far as a student's car goes, the legal guideline is if it is "parked on or adjacent to school property". Some judges define adjacent as being anywhere within sight of the school. Others have likened searching any property of a student's to crossing a border, where there is implied consent for search because the student has chosen to bring property onto campus.
Even though they are on school property and are minors, students do have some right to privacy and to be free of unreasonable searches and seizures. What is really different is that there is no need for a warrant in school searches and the right to privacy has to be balanced against the school's duty to provide a safe environment for students.
Perhaps the most important case in this regard is New Jersey v. T.L.O. In this case, the Court held the school officials may search a student's person or property as long as they have a reasonable cause to do so. The Court in that case took a relatively expansive view of what is reasonable.
So, students do have privacy rights, but these are not as strong as the privacy rights of adults or even those of minors outside the school setting.