Well if the horse isnot dead yet you can contact a wild life sanctuary and they provide a vet to put thehorse down without drugs. Here they shoot it on thehead but it's not a bullet. They transport the horse carcass to the sanctuary and deal with it there. A horse will feed a lot of animals for a good while. We did this with a real old dear mare and made all the arrangements ahead of time so when she could not get up recalled them.
A number of ways, transport the carcass to the state lab for a necropsy and they will dispose of it. If you have enough land, use a backhoe to dig the hole. Call a place that specializes in horse disposal or in many state you can haul it to the landfill