How far apart your hands are holding the split reins depends on what you are doing. If you are trying to get them to collect and drop their head, your hands will be farther apart while you "pulse" pressure in the side-to-side fashion on the bit. If you are asking them to stop, you will have your hands closer together and near your pommel. If you are teaching a spin, one hand will be up and one hand will be out fairly far. And so on....
Yes, the extra reins hang down the side of the horse. The rein that is attached to the left side of the bit, will cross over their neck and hang down the right side. Vice versa for the other rein. Make sure the reins aren't too long for your horse in that he could possibly step on the ends of them.
When direct reining a young horse, I hold the reins with my palms down. I use my fingers to apply pressure to my rein and don't need to use my entire hand very often. To quickly adjust my hand position, you just slide the reins through your hands. So if I want my hands closer together and the reins shorter, I let the ends of the reins slide through my hands as I move my hands inward (while hanging on firmly to the part of the rein that goes to the snaffle bit). Takes a bit of practice, but you can do it quickly and without thought once you get the hang of it.
Hand position depends on what you are doing. In general, though, your hands should stay near the pommel of your saddle. But of course, this can change based on what you are doing.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.