Last night, we put two of our cats into the coop. One made it out within hours. The other was out the next morning. Clearly, there are places where skunks and raccoons could get in if the dogs didn't keep them away.
We keep our feed off the ground by about ten inches. Low enough for the hens to reach and high enough to discourage mice from finding it. We haven't seen much rodent activity in our coop/run.
BTW, our coop/run is 1500 square feet, so a cat wouldn't feel cramped in there, but we agree that the cat needs a way to get away from the chickens when necessary.
As far as cat-chicken fights, what I read is that neither really wants to tangle with the other, unless the cat were starving, which we prevent. In that scenario, the cat has the advantage against one chicken, but with two roosters, the cat probably wouldn't try anything, certainly not just for fun. If there is easier prey and plenty of cat food, the cat should leave the chickens alone.