Most of them do. They're usually constructed as restrictions on the government's authority rather than grants of rights to a specific group of people, and there are court rulings clearly stating rights like freedom of speech and freedom from unreasonable search apply to any person subject to the government's authority.

Non-citizens don't get the right to bear arms as far as I know.

This *1000. It's a supreme failure on our educational system and society at large that most Americans don't think about our government and our rights in these terms.

The Constitution actually takes away our rights. It has to, it lays out the powers the government has over us. Even the bill of rights doesn't technically give us rights---it explicitly details the ways our government cannot operate.