Don't worry about chicken manure. If they have good feed and enough water, it comes out liquid most of the time. It just soaks into the ground or bedding. The only time I've seen it get solid is when feeding corn, which is probably the worst feed nutrient-wise (it's used because its cheap for the number of calories, but really try to avoid it).
There will be nothing there for the dogs to eat, it just soaks in. However, many dogs will still be able to smell that spot and will roll around in it.
As for chickens eating dog droppings, they won't if they have something better to eat. They'll scratch them around and help break it up.
The only transferrable pest to worry about with dogs and chickens are tapeworms. If one of your chickens gets one, you'll know almost immediately, they die very soon after. It's almost a non-issue given how quickly a chicken will succumb to it. They lose weight and the other chickens will kill the infected one. It's pretty rare, I wouldn't worry about it. Only an issue in the worst possible living conditions, not the type of conditions you would have with a backyard flock. The type of tapeworm that lives in chickens doesn't live in human hosts, so no worry there.
When it comes to free-ranging with dogs, don't. You'll lose a few birds before getting the dogs trained not to chew on their heads. It can be done, but it's a pain to deal with. That run is the dog's territory, the chickens are intruders. The dog will defend it's turf. Even if it doesn't, big flappy feathery things that let you chase them... too much temptation for most dogs. In a large area, they'll learn to ignore each other. If the dogs are sharing a quarter acre or more with the chickens, training will work. In a small confined run, it will be carnage. Not sure how big your dog yard is, or the temperament of the dogs. You'll need to judge that for yourself.
For tractoring, clean the area of dog droppings before moving the chickens in. That takes care of that issue. The area the chickens leave behind will be mostly clean if you move them regularly.
I've found chickens are easiest to manage in a coop. It's easy to get the eggs, all the manure is in one place to be composted, feed is always covered, water is always available, light and temperatures are controllable, and predators are a non-issue. It's just a lot easier to control the conditions. You can build a run for the chickens outside of their coop, but keep them confined to a single area. You can always toss grass clippings into their run as bedding and something for them to eat / search for bugs in... It's easier to bring the pasture to them.
