Growing up on a horse breeding & training farm, I have much experience with horse penises. Wait! That sounds real odd… I didn’t mean mean it like that!
Anyway, on a typical adult stallion (excluding draft horse breeds), the penis is about 2.5-3 inches in diameter, and length is about 12-18 inches. Think of a typical large kielbasa or similar sausage in the grocery store. We never measured any, actually, it never mattered; size varied by stallion, but all seemed sufficient to effectively breed a mare. It’s large compared to humans, but seems to me to be pretty much in proportion to their body size.
The head end of a stallion’s penis is basically flat, rather than rounded like on a human. During & after ejaculation, this swells outward, and can get to 4-6 inches across, about the width of a typical saucer. I presume the purpose of this swelling is to block the passage, so that the stallion’s sperm moves in the right direction up towards the mare’s egg. There isn’t any kind of foreskin on it, like on humans. Probably not needed, since the penis is normally held inside the body.
Breeding in horses is very fast. Typically it’s “done in 60 seconds”. But it’s commonly repeated 2-3 times a day for the 5 days or so a mare is in heat.
There is a fair amount of what you might consider ‘foreplay’, but that’s mostly done by the stallion to verify that the mare is ready to breed. Experienced stallions are cautious, because a mare that’s not in heat will reject a stallion quite violently, kicking, biting, etc. And the stallion is in a vulnerable position, and can easily be seriously injured by that if they aren’t careful. Also, young mares, even when in heat, often don’t know how to respond appropriately.
So it’s quite common on breeding farms to put kicking straps or other restraints on young mares. And they often have a “breeding stall” on the farm, specially built for this, with padding, etc. to prevent injuries to either the mare or stallion during breeding. Breeding is typically done with 2-3 humans leading the horses, and making sure it all works right.
Remember that in horse breeding, it’s common to cross breed to different bloodlines. So the horses are typically strangers who have never met before, and the mare has usually been transported to a strange location, away from all her normal companions. And they are both very valuable animals, especially the stallions.
There is quite a bit of semen ejaculated by a stallion. I’d guess about 1/3 of a cup or so. Way more than is needed to impregnate a female (just like in humans). In Articifial Insemination, semen is captured and frozen into ‘straws’ for use in later breedings. It’s common to get a few dozen straws from a single ejaculation. These can be frozen and held for years before being used to breed a mare. In breeds that allow AI, it’s not unusual to have new foals born from a stallion that has been dead for several years.
The stallion’s penis is normally held inside the body, except when urinating or breeding. (Stallions can urinate with the penis inside, but normally don’t.) Thus it can get rather dirty, and covered with an ear-wax like substance, which can get up to 1/2 inch thick on the penis. So one job on a breeding operation is washing the stallion’s penis. Yes, really!
Washing is commonly done both before breeding to prevent any kind of infection in the mare (which would be very dangerous, usually fatal to the developing foal). And commonly done again after breeding, to prevent anything from the mare from infecting the stallion. (Even though most breeding farms require a vet’s inspection of the mare first, to verify that she is not infected.)
Stallions often object to this washing at first, unless you start when they are young. But they soon learn to accept it, and will even start to get erect during it. Partly just because it feels good, but also because they are smart enough to learn that this means they are soon going to breed a mare. Often breeding farms have a special halter & lead that they use on the stallion only when breeding. He will soon learn to recognize this halter, and will get quite eager as soon as you put it on him. Horses aren’t dumb!