Chickens do make a bit of a mess but less than dogs and I do get some mice in the run eating the food when I leave the run door open while they are free ranging. Perhaps if you keep the run shut up with hardware cloth on it and under it and leave water outside for the chickens you won't get mice, but I like to leave food out for the girls during the day as well.
Somebody else will have better knowledge of what breed of chicken is nice and friendly but I hear Buff Orpingtons and Delawares are. I am happy with my EEs but I have heard they are not the best layers. They're working well for me but I don't eat a lot of eggs.