How does the horse compost dog compost?

Unfortunately, no. Its chemical composition and likelihood of carrying various parasites makes it slow to decompose and a possible source of infection.

Is compost toxic to dogs? (AKA "Why am I supposed to fence off my compost pile?") Why do dogs eat dead, decaying vegetable matter? Compost can be toxic to Sherlock, and while you may not think that...

We scoop it into a manure spreader and spread it across the fields before the hay comes in or between cuttings.

No. Carnivore manures are not good for compost because they do not compost completely and may contain dangerous parasites.

Anything organic can be composted, but certain things perhaps should be left out of the compost heap. Meat scraps, for example, will degrade in time, but can create an awful smell while they do it....