That non-halter thingy is a mouth speculum.
Besides the vet having their arm in the horse's mouth, it is VERY dangerous for the person restraining the horse to have an insufficiently sedated horse swinging the heavy metal speculum around. It would be very easy for them to knock you out or worse -- it's bad enough being hit by their skull, let alone a metal implement!
Sedating horses for dentals is one of the great debates among horsepeople, as non-vet dentists do a lot of floating in many states but are not able to sedate other people's horses. Leaving aside the dentist vs. vet debate, the arguments for floating without sedation come down to lower cost, avoiding side effects of sedation (with exceptions such as GeeQued's horse, sedation produces no significant adverse effects in something like 99.99% of horses*) and speed. The arguments for sedating the horse include safety and the fact that many horses do not allow power floating without sedation; power floating is easier on the practitioner and allows the vet to do a better job of grinding down the very last molars -- it is hard (I won't say impossible
😉 ) to rasp down the caudal edge of the last lower molar because there is no room to move a hand tool back and forth.
*I have personally witnessed thousands of sedation events without ever seeing a horse colic, collapse, have an allergic reaction, or anything else; the equine vets I know in private practice sedate multiple horses a day and go YEARS without seeing adverse events.
More on heads:
When holding a horse who might try to swing their head, I try to stand out of the arc that their head naturally makes side to side so that they don't hit me with their head. If that sentence didn't make sense, think of the horse's neck as the radius of a circle and the horse's head as traveling along that circle. You want to be either inside the circle -- a little bit behind the head, next to the neck -- or outside the circle, not exactly on that circle.
Foals are generally restrained by standing on the opposite side from the vet. If it's just a physical exam, just rest one hand in front of their chest to keep them from moving forward and gently wrap the other around their hindquarters to keep them from moving backward or wiggling sideways. If the vet is taking blood or doing something similarly objectionable, your "battle stance" involves one hand on the base of the tail pulling the tail upward (controls the hind end and sometimes has a calming effect) and the other hand goes under their jaw and grasps the ear, so that the bottom of their jaw is sort of resting in the crook of your elbow. This allows you to position and stabilize the head to the vet's desire. If you are restraining a foal, be warned that the little buggers like to throw their head up in the air if upset and can whack you pretty good on the chin with their heads. Either keep your head back so that you are not in the line of fire, or lean over them so that your chin is on the other side of their midline. Restraining foals is a little like playing rugby, only the other side doesn't follow the rules.
On feet:
GeeQued gave the basic approach for a well trained horse. It helps sometimes to start at the elbow/stifle and run your hand down the whole leg -- they know what's coming and will sometimes pick it up right away, although that can be less helpful and more annoying if you are trying to palpate the leg. If that doesn't work, you can grasp the pastern and lean into the horse with your shoulder. On a big draft horse with lots of feathering, I've seen people lift the feet up by the hair on the fetlock. If it's a front foot and the horse doesn't want to take weight off that foot, have the person holding the head (if you have one) move the horse's head away from you.
General horse restraint:
You've probably already been told this, but (unless instructed otherwise) when holding adult horses always stand on the same side as the vet, a little in front of the horse and a little to the side, facing the vet. By being a little in front and facing backward, you can see the whole horse, including their head, so you can pick up on cues that the horse may be about to react, such as ears pinned back or flicking back and forth, angry look in their eyes, etc. You are on the same side as the vet so that you can see what the vet is doing (and predict how the horse will react) and so that if the horse decides to try to kick you can yank their head towards you, which swings their butt end away from you and the vet.
That's all (ha! a novel!) for tonight. If you have more questions, I'll try to answer them tomorrow.
