I just watched the first little bit but the horse is in schooling stage still. He would naturally poke his head up in the air & is only starting to learn to put it down with leg & hand aids. Once he has learned this, she will be able to ride him with less pressure (or she could decide to be a happy hacker & just let him put his head wherever he wants & ride him on a loose rein-not my advice though)
I don't know much about western riding. But she looks to me like she's doing dressage in a western saddle. I didn't know that western riders held their reins like that. Her hands do look a bit unforgiving, especially at the canter, where you would want your elbows to move with the motion of the horse. But not knowing much about the style of riding, I couldn't say for sure.
Not typically. When doing eqitation you need just light contact so you always have a connection with your horse, but a good hacking horse should be able to be ridden with a loose rein. If you need a tight rein perhaps the horse needs either a different bit or is not properly trained. Consult with a local horse person. Good luck!
There were more things wrong in that video than just the reins. The rein length would really depend on what they are training for. I rode/showed Saddlebred saddleseat and your reins are not droppy like pleasure classes. Trying to sit that trot - looks very uncomfortable for the horse - even with the western saddle you can post to the trot.