Do horses really not mind have pieces of metal NAILED to their feet? How does one get the horse to accept this without going nuts when you start hammering? I understand that it doesn’t hurt since the hoof is a really just a big toenail. Do the horses like having the shoes on? Do they have trouble getting used to it?
Many things we ask of horses are unnatural. This is just one more.
Yes, they really don’t, if it’s done properly.
Slow introduction and training is the best method. (Start out just tapping on their feet and work up to actual shoeing. Doesn’t take long.)
True.
Who knows. Do you like wearing shoes? I don’t, but they make me anyway.
Not generally.
Did you know that there are actually “orthopedic” horseshoes of various kinds, to help treat/prevent different conditions and diseases?
IANAHorse, but I did work for horseshoers for several years.
“Shoeing a Horse”
Just wave your hands and arms, while loudly repeating, “Shoo horse.”, usually works for me. 
To expend a bit on what redtail23 said - I started picking up my horse’s feet and cleaning them and tapping with a brush when he was a yearling, long before his first pair of shoes. He soon got used to it. He does get fidgetty, but that’s in part because his farrier when he was young had emphysema. He’d do two shoes, stop and smoke a cigarette and catch his breath, then to the last two. Irish got used to taking a break, too. Irish is very tender-footed, and if you walk him on gravel when he isn’t wearing shoes he’s like “oochy-ouchy-oochy”. Plus, he’s a thoroughbred, which have notoriously bad feet. Although his aren’t too bad for his breed, he’ll get hoof cracks if his feet aren’t attended to.
StG
Which rather leads into the question, how did horses manage in the wild before people started managing their hooves? Or is it just that some breeds have become so domesticated through breeding that they couldn’t survive in the wild any longer?
Projammer - Horses that are not feral live much longer lives than the mustangs. Also, through survival of the fittest, mustangs tend to have very hard hooves. Mustangs wear their hooves down on rocks and hard ground. But with feral horses, if they do have a hoof problem, it can be the end of them, because they can’t run as fast from predators.Thoroughbreds and been specifically bred for centuries for one thing - speed. Their bones are lighter and more fragile (less weight to move). Their feet aren’t great, because the fastest horse doesn’t necessarily have the best feet, and it’s the fastest horse that’s bred. TBs tend to be something of sprinters, while Arabs tend to be better endurance horses.
Horses tend to have a hard time foaling, too. Those long, long legs make it difficult unless the foal is in just the right position. My vet used to say “cows, women, they’re built to give birth. Horses need to be babied.” As you can imagine, with feral horses, those who can’t give birth readily die very quickly, taking those genes with them. If you look at native equine species, they tend to be more compact, not as leggy.
Teeth can also be an issue. Many domestic horses have their teeth “floated” once a year or even more often. The dentist files away sharp edges and spurs that can interfer with grinding the grain and hay properly for digestion. I’d imagine that feral horses die younger because their teeth grow in ways that make it painful to chew and sooner or later they waste away, growing weaker until a predator gets them.
StG
In the wild, horses evolved on dry prairie, and the constant wearing of their hooves on turf with hard dirt under it served to keep them worn down at the proper length, much like a dog’s toenails.
Also, wild horses were not asked to carry 150-200 pound humans on their backs, either. Combine the additional weight with something like pavement and you get far more wear and tear on their feet than what their wild ancestors did.
IME horses don’t have trouble with the nailing per se, but many have trouble with standing still for the required time. Some are just kind of fidgety (see StGermain’s post) because they don’t know any better. Discomfort in the legs or back can be a problem - older or injured horses may find it uncomfortable to bear weight on three legs while the fourth is being worked on, and holding up a leg (esp. a hind leg) can get uncomfortable.
Hot shoeing presents its own problems. Burning the shoe on doesn’t hurt, but some horses don’t appreciate the smell of burning hoof or the wisps of smoke that float upwards.
Also, domestic horses don’t have to run away from predators and rival horse gangs.