I own horses and I've been around them most of my life. Horses would not be laying down in their own manure or urine if humans didn't keep them in such confined spaces like stalls, or small paddocks, AND not clean those spaces often enough.
Some horses are similar to donkeys in that they will poop in the same spot, somewhat, and it will be their latrine.
Blame the owners or just the fact that many people can't afford big pastures.
A stud's penis is cleaned before he breeds a mare. An attentive horse owner will clean a horses penis if it gets too much smegma on it (flakes), regardless if it is a breeding stud or a gelding. I'm sure the stud that would possibly breed her could have his penis washed an hour ahead or two before he meets your wife for amorous times.
What really needs to happen aside from a washed cock:
The horse should be gentle or well mannered enough. The owner should display some know-how and lots of comfort around horses, and the ability to anticipate accidents.
Your wife has some experience around barns and horses, that should go a long way. People that are new to horses don't know.... what they don't know.
She should be able to accommodate lots of size vaginally, like a 4" width inside past her vaginal opening, if the stud is a large pony or full size horse. The flare actually can be 5" on a 1,000 lb horse, but the outer edge of it is spongy and compresses a little. If her vagina is just 5 or 6 inches deep, like some women, definitely only try a mini. Even with a vagina that is 8" to 9" deep, there is still a lot of deep pressure. She should be familiar with having that limit pushed, but that probably goes without saying.
A mini or shetland pony, or a taller pony that has a smaller cock, would be the best way to have some first experiences.
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I never see it in vids or pics, but a very short bench placed beside the stallion. Place the flat bench right to the left side of the stud. She can bend over and hold the front of the bench and have her right knee on the bench, but the left foot down on the ground to be ready to get up and step away quick. This would be with a more gentle stallion that wouldn't stop his front hooves a lot or pace side to side.