Yes, some can carry mites but I don't think they can live on humans. I use "food grade Diatomaceous Earth" (DE) to get rid of mites, can be found at pet stores I believe, probably farm supply stores and other stores too. It is non-toxic, there are other things to use but I forget what they are called at the moment (will look it up). You just dust the pigeon with the DE and the dust punctures the mites bodies but doesn't hurt the bird.
That wildlife organization sounds good, just make sure they take pigeons cause some won't since pigeons are non-native
Yes you can put her in a larger cage.
(Sorry it took so long to respond)