For the mosquitoes: They can carry chicken pox and other nasty things. My flock has chicken pox right now and it's no fun at all. :\ Eliminate any standing water. I've heard of people putting goldfish in waterers. but it only works for certain types and I'm not sure how it would work. (Though I've been meaning to look into it, 'cause we have a lot of mosquitoes around here.) You can put apple cider vinegar in the water. It's good for the chickens and is a mosquito preventative.
For the mites: My chickens got mites a sometime last fall. I bought Permethrin (spray) from the local feed store and sprayed it on them every day until it cleared up. You can use food grade Diatomaceous earth (DE) in their nesting boxes, which helps prevent them, and it's pretty cheap. Do your chickens have a place to dust bathe? If not, they need one. That's why mine got them. I had them locked up and they had nowhere to dust bathe. As soon as the started free ranging they went to the manure pile and started rolling around in it. It's their natural why of preventing mites and lice.
ETA: They don't need a manure pile to dust bathe in. I just realized how that sounded. If they have access to the ground they will make a spot to dust bathe. All they need is dirt.