Re: coyotes w/mange?
I leave em lay, the mange epidemic here has for the most part run its course. Occasionally we run into an area with it, but its somewhat of a rarity (most years). But than again, up here mangy ones dont last long once the snow flies.
If your concerned about removing them, id handle them in "non-infected" areas, or with gloves(if the tail is mangy, dont go rubbing and stroking it, etc). Wash your hands and/or shower well as soon as possible after handling infected animals.
Ive handled lots of infected ones and the only real precaution ive used it not getting grabby with infected areas of the animal.
If you can, burying infected carcasses in a fashion where they wouldnt get dug up would be ideal.