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If it is a fresh kill, I believe your dog can catch scabes/mange from a coyote, just as other coyotes in the pack could. I wouldn't worry about an old corpse though.
The mites that cause mange in animals will also cause scabies in humans. Same type of mites, different name for the species affected. So you are also in danger. Don't hug the dead coyote I guess?
Talked with a guy that got mange from carrying a coyote, this was in the ealy 90`s. Said it took quite a while to get rid of it, didn`t want to go through that again. I make sure and get rid of any mangy kritters that are shot near the farm.
Most people just let mangy coyotes "recycle" where they fall, if you must dispose of one be careful. I think disposable gloves and a garbage bag would be prudent if you must handle mangy coyote carcasses. Some diesel fuel and a match works well too if you are in an appropriate location.