One or more of the hens may be plucking and eating the beard feathers. My RIR ate the beard off one of my EEs on two separate occasions. Others have mentioned this as well.
Chickens should always have water available during the day. Check out pet stores for bowls that can be attached to the wire. There are some made for dog crates and also for bird cages that may work for you.
Doves can be so hard to sex. That does seem more like a female behavior but I have seen my males do it too. Is this your only dove? You can offer a small basket or box as a nest, and you may get eggs. That's the only way to know for sure, unless you have a pair, and see mating. Or if you have...
It can take a few days or a few weeks. I wait until the flock are not paying any attention to the new girl. When you let her out into the run, scatter some scratch or sunflower seeds to distract them all. Chasing her is normal, but if they are constantly hounding her, jumping on her, Pecking to...
Lots of times animals are so nervous in the shelter that they don't show as friendly, especially cats. How nice that she has relaxed with you and it sounds like you will be happy together.
There will always be times when you will want to confine them, even if those times are rare. Better to have it and not need it than to need it and not have it.