Actually, that is completely up to the airline. Your right to carry a weapon ends the moment you enter that airplane. It's private property, and the owner has the right NOT to allow you on board with a weapon if they so decide.
Also, in spite of the Second, I don't have to allow anyone to carry a weapon into my home, unless they have a warrant.
No,it's not. The FAA are the ones who decide this. Just like they are the ones who decided there would be no smoking on ANY flights.