We drop our reins on the neck, simply because we're used to doing so for jousting. In a joust, you hold the reins long enough to come about and lay on, but must drop them as you set the lance and present your shield--even if the reins are long enough for the horse to be comfortable when you raise your hands like that, you run the risk of breaking the animal's jaw if you're unhorsed on impact, especially in heavy armor jousting, when you and your equipment likely weigh in excess of 300 pounds. Because we do both, our reins are short enough to be dropped at the neck without running much risk of getting underfoot. That said, there's absolutely no reason (that I can think of) not to secure the reins to the saddle or elsewhere prior to doing a mounted archery run, since you don't need to turn around in the list prior to starting the track. In fact, it's a very good idea--our troupe just hasn't implemented it thus far.
I've personally seen a few different set-ups for a mounted archery track--one involves going all the way around a track along the rail of an arena, shooting half a dozen or more targets at different heights and angles. The Kassai course is a simple ninety-nine meter list, with a single target that rotates during the run to face the rider. The director of our troupe set up a course along some of the trails on his property with targets at various heights after seeing Brave. That one was fun, but a serious challenge, because the targets were also on both sides of the path. That complicates things significantly--it's REALLY hard to move the bow back and forth across the horse's withers, awkward to twist toward your draw hand, and I don't think I would ever want to do it with people watching, because I only hit about a quarter of the targets.