I asked about horse breading and training, and /u/FlyingChange gave me this great and detailed answer which spills over into some of your questions.

1. Thanks for linking over to my other answer, /u/JustMe8.

2. This is an interesting question. Medieval Europeans warhorses were bred mostly by the people who would be riding them- nobility, knights, etc, and the training thereof would be within the realm of the nobleman and his staff.

Now, it is important to understand that "breeds" as we understand them didn't really exist at this point in time. Breeds came more as national identity developed in Europe (which is why we have breeds like the Hanoverian, Westphalian, Dutch Warmblood, Trakehner, Andalusian, Lippizaner, etc.).

Instead, at the time, horse types were more of classifications for use. There were three types used in combat: destriers, coursers, and rounceys.

A Destrier was the biggest, strongest, and rarest type of warhorse. They were also expensive, which is why many poorer knights and men-at-arms would use other types of horses. But, a well off knight would ride a destrier. The destrier was specifically bred for mounted combat and for the traits that would make him successful as a charger. This included size, strength, and large, powerful hindquarters (which is still a desirable trait today in many working horses). The destrier would typically only be used in a battle or a joust, and the knight would use a riding horse for general use.

The courser was a slightly smaller, faster, and lighter horse that was still used commonly in battle. They were less expensive than a destrier and were generally more useful as an all-around horse. If the destrier is an armored vehicle, then the courser is a Humvee.

And finally is the little rouncey. A rouncey was simply an all around horse with nothing terribly exiting going for him. A poor knight or man-at-arms would ride him in combat, around town, and pretty much wherever.

So, to answer your question, they were pretty darn successful. With that in mind, the breeding programs were not super precise. They would find a good stallion, breed it to a good mare, and then horse for a good baby. The concept of a studbook (a book detailing stallions, mares, and their offspring), a pedigree, and papers detailing that pedigree that followed the horse from owner to owner came from the Arabian people, who have kept amazingly detailed records of horse breeding for centuries. Europeans didn't really start doing that until trade for Arabians and Barbs picked up kind of in the tail end of the Renaissance, which is when we start to see specialized breeds with specialized uses. But, even still, we kind of still have classifications like they did in the Middle Ages- Cold bloods are big draught horses, hot blooded horses are smaller, faster, more energetic horses, and warmbloods are a cross of the two (which are big, but still energetic and swift).

Fun fact- most horses at the time were very small- often around 14 to 15 hands high. To put that int perspective, modern rules states that a horse under 14.2 hands is a pony, and the average riding horse is 15.2 to 16.2hh. We know this by looking at horse armor and by looking at the dimensions of horse transport ships.

Any follow up questions?

Sources:

Medieval Warfare: A History, Edited by Maurice Keen.

The Culture of the Horse: Status, Discipline, and Identity in the Early Modern World, edited by Karen Raber and Treva j. Tucker.

A Knight and his Horse by Ewart Oakeshott.