Do states have to abide by the U.S. Constitution?

Of course they do. If the states didn't follow it, then NO ONE would be following it.

Public schools cannot endorse a religion. States cannot endorse a religion, thanks to the First Amendment.

States must adhere to the requirements of the Second Amendment as interpreted by the Supreme Court.

The provision of the Constitution that ensures this is called the Supremacy Clause. Look that up. The states cannot act contrary to the Constitution. They can make laws about topics on which the document is silent, but if a state law and federal law are in conflict, the federal law wins. Also, the 14th Amendment incorporates the protections of the Bill of Rights to the states.

The Constitution is the supreme law of the land, so the states must abide by it.

A public school cannot endorse a religion; a private school can.

Your friend is wrong. He should read the incorporation clause of the 14th Amendment.

A public school cannot endorse a religion.

This is mostly guesswork, but I'm pretty sure they have to.

And I don't think they can endorse a religion (unless it's a Catholic school or something of that sort) because that would be religious intolerance since it would go against other religions--including atheistic views.