First of all, Vermin can be any animal that is destructive, annoying, or injurious to health. So yes, pigeons can be vermin.
Secondly i would not suggest eating the pigeons that you find in the street, however you have killed them, as they have little meat on them and the meat thats there will be very stringy and tough.
Thirdly, it is illegal to kill any kind of dive, and only ferral pigeons (street pigeons) can be killed to control population.
And finally, to answer P.Greenwood's question. The pigeons deformed claws are usually caused when the bird gets their claw stuck, and they try and fly away intil the claw is pulled off. If this does not work, they will try to peck their claw off.