I have had no problems with keeping pigeons and chickens together. I have two coups which they have their choice of. One is equipped with special pigeon perches, the other just has the basic chicken perches. My pen is 40x20 and each coup is 10x10. I haven't had problems with either bird getting sick, nor have i had problems with my roosters attacking and injuring the pigeons.
The key is space and not overcrowding them. Pigeons are weaker and have no defenses against a large rooster and both pigeons and chickens maintain territories. Having a large enough space will help to prevent disputes over territory and places to sleep. Be sure there are places above the ground where the pigeons can get away if they need to. I've found that my pigeons and chickens don't even bother each other unless it's time to eat.
Pigeons and chickens can eat the same foods without much trouble. All of my birds, even the quail, are fed a 16% protein all purpose poultry ration, both Albers and Payback carry a similar ration. Layer pellets and scratch grains don't hurt either, since occasionally I am forced to switch feeds. I have not had a problem, even with breeding pairs when I feed these rations.
Be sure to feed in a large enough area that everyone can get enough to eat. The chickens tend to push the pigeons away from the feed if there isn't enough room. Feed in two different places to avoid problems.
Anything else you want to know.