Key Steps to Managing Horse Pastures 

Yes. Mowing will not always disperse the manure. You can read more about pasture management at extension.umn.edu/horse-pastures-and-facilities/managing-established-horse-pastures

I'm not sure research exists to answer your question. However, because horses tend to exhibit preference when grazing, we suggest grazing them first, followed by goats, and then chickens. The goats will hopefully consume the forage horses find less palatable and the chickens should help spread out manure piles. We are not experts on pig grazing, but pigs tend to "dig" and can cause significant damage to pastures. Therefore, it's not commonly recommended to graze pigs.