Children have the right to live free from hunger, abuse, neglect and other inhumane conditions. And children have a constitutionally-based liberty interest in the protection of their established families. However, a parent's right to autonomy in child rearing is a fundamental right protected by the Fourteenth Amendment. It is the government's assumption that parents have the best interests of their children in mind when they make decisions for them.
One of the decisions parents make is to allow schools to "parent" their children during the school day - and to make the necessary decisions for these children during that time.
There have been cases where children have won court battles in the past, certainly - especially where the cases have involved the severing of parental ties. But these children have only been able to bring these cases to court and win these cases in court with the support of adults who joined them in bringing their cases to court.
The U.S. Supreme Court is increasingly less likely today to support any child's claim to constitutional rights. In the liberal Warren court, 100% of the "child constitutional rights" cases brought before the Court upheld children's claims of constitutional rights, whereas only 22% of the constitutional cases brought before the Rehnquist Court were decided in a child's favor. The cases heard in recent years have included equal protection, privacy, free exercise of religion, due process and free expression.
Children do not share in an adult's Constitutional rights. Children may be detained in situations where an adult would not be, for instance. Bail is not set for children. Children do not receive the benefit of a jury by their peers. These are only some of the ways that children's rights differ from the rights of adults. Those making these constitutional rights decisions have increasingly held that children are not capable of making certain decisions on their own (children need to obtain parental consent to have an abortion and parents can make medical decisions for their child, as two examples), and they are not capable of handling all of the legal rights guaranteed to adults under the Constitution.
So, again I will say that the correct answer to fancypants' question is not "no" and is not "sometimes" but it is "YES - a police officer can search a student, without the student's consent, if the student has not been arrested." Not only on school property, but off school property as well. The "gun situation" is definitely a fair response.