Good questions OP! I’m an equestrian and grew up idolising CDJ, and I’m so disappointed and upset to see her treat a horse this way, but I understand that to a non-equestrian it surely doesn’t look any worse than what happens in racing.
So to address the whipping issue first. In racing, they use padded batons and have a strict limit on how many times they can make contact with the horse before the jockey will face consequences like being banned or suspended. Most of the time the jockeys are basically waving the whips around in the air. You’ll see similar sorts of whips used in other jumping sports too — the cross country phase of eventing, showjumping, etc, and again their use is heavily monitored in competition.
In the video of CDJ, she’s using a lunging whip, which has a long “stick” bit and an even longer string/rope off the end. They’re designed to be used when “lunging”, which is when the handler is standing in the center of a circle with the horse moving around them. Basically you’re supposed to use whips like these by giving them a little wiggle on the ground behind and away from the horse in addition to voice commands to make the horse move a little faster. The string part will move and make the horse move away from it. It should never ever make contact with the horse under any circumstances.
For reference, there’s another type of whip called a schooling whip, which is thin and fairly long and usually has a very small bit of string at the end. These are designed to be used while riding, and basically to give the horse a little tickle behind your leg to get them to move off a little faster if they’re ignoring you squeezing with your lower leg. (This is really simplified but should give you an overview).
So the reason this is all a big drama is that CDJ was always respected for being kind to her horses, for loving them, and for reaching the very top without resorting to some of the cruel practices that you do (sadly) see in dressage and other equestrian sports. She was supposed to be one of the good ones. So to see her following a horse around an arena continuously whipping it with a lunge whip to the point of the horse clearly being distressed and scared, and her apparently just going about her usual business, is a huge problem for a sport that is already under scrutiny for the unscrupulous actions of a few really nasty types. It’s heartbreaking to see a horse in pain and fear, and it’s heartbreaking to get a peek behind the curtain at the practices of someone who was heralded as being the kind who puts her horses before her sport and who is/was held on a pedestal by so many equestrians.
It sadly brings all of equestrianism into a really bad light. Dressage is supposed to be about the partnership between horse and rider — about their bond and connection and harmony. Unfortunately that “harmony” can be faked through fear and aggression, and now someone we all thought wouldn’t resort to such tactics has been shown to do exactly that. Most equestrians I know (myself included) would die for their horses, and it’s a real blow to see one of our best known representatives acting like this with apparently not a care in the world.
Hope this helps clarify the situation for you!