Well, one thing I know is you don't want to pick up the head of a gaited horse in order to get its hind end under it. You want to encourage her to lower her head, which in turn will help her lengthen her stride. When she lengthens her stride, her hind end will reach under her and you will have much more of a contained position than with her head up and back hollow. Usually when you just try to teach her to get into a more collected position by lifting her head up like that, and keeping such a tight rein, the only thing you're going to get is a hollow horse with a shorter stride. So I would first ignore the gait and teach her to lower her head. Worry about the gait later.
My mare did that "hopping" thing once, too, so I know exactly what you mean. It's cross firing, which is when a horse half canters and half running-walks, and it feels like she's consistantly tripping a little or something, but you know she's not. It just takes some time and a lot of consistant riding to get rid of it. Every time your dad's horse does that "hopping" thing, take her down to a walk and then cue her back up to a running-walk and do that over and over as many times as nessesary. Then when your horse keeps herself in the running-walk for 6-8 seconds or so (at first), let her stop and rest and let her know she did the right thing. Backing up and then going straight into a running-walk also helps because backing up helps to collect them.
It appears that your dad is keeping a very tight rein on her, and that is most likely the reason she's doing that; she can't do her running-walk as easily as she can canter when she has her head all drawn up with no room to "head bob". Riding a gaited horse is a whole different ballgame than riding the non-gaited ones, isn't it!