no but judges do not like it if a boxer hits a non boxer
It depends on the jurisdiction.
Penal code determines this kind of issue. Each state has its own penal code, so if the law exist in one state but not in another, you could see how the ones in the second state would assume its an urban legend when they find out their state doesn't have this law.
In Puerto Rico boxers fists are considered ''arma blanca'' which is essentially a weapon. So a registered boxer punching you could get the same sentence as a man hitting you with a bat. It's assault with a weapon.
No, that is an urban legend that martial artists in general have to register themselves as lethal weapons.
No, it's a given once they are registered or licensed as a professional fighter.