Who were the first veterinarians?

When did someone decided to treat livestock for diseases instead of culling them?
Do the still shoot lame horses?

No one would shoot a horse to put it down except in an extreme, extreme emergency when no vet could be summoned. Usually they are euthanized, just like a cat or dog, but on a larger scale. In James Herriot’s books about being a large animal vet in England from around 1920-1960, he talks about using something called a “humane killer” which I think was a kind of bolt gun. However, a close friend of mine is an equine vet, and that type of thing is not used today to the best of my knowledge.

“Lame” can mean a variety of problems both cureable and uncurable. It is very rare to put a horse down for lameness alone except where the lameness causes the animal continuous pain (as in exteme cases of Founder or Navicular syndrome), or the leg is traumatized to where it can’t bear any weight (such as a broken bone). It’s not easy to immobilize a horse for 6 weeks. I can think of a couple of famous racehorses, like Alydar and Ruffian, that were put down after serious leg injuries, although heroic efforts were put forth to save them.

Most horses that are too lame to work are sent to a retirement farm or otherwise retired to pasture, or if they’re less lucky, slaughtered for meat (horsemeat is popular in Europe).

I would imagine the first vets were the breeders or caretakers of the animals when humans began keeping livestock instead of just hunting what they needed. They would have learned what plants to avoid by watching reactions in the animals under their care, and learned the hard way how to save injured animals.

Sometimes even today an injured animal is saved, even if no longer capable of racing, jumping, etc., in order to breed it.