Pigeons are not game birds, there are no seasons or bag limits on them, they are considered feral, most farmers would appreciate tem being shot. The only law on the books regarding pigeons is a law that says you may not shoot homing pigeons. How one could tell if a bird was a homer while flying eludes me. As far as eating them, I have no problem eating barn pigeons, especially the squabs, the old birds are tough but can be used for pot pie. I would not eat pigeons that came from a city.
When you are up to your butt in alligators, it is hard to remember your intent was to drain the swamp. Stay focused!