It depends, In English, you ride very close contact with your leg on the horse. But I've seen Western riders who ride more straight legged with their lower leg totally off the horse. I had one horse, the black Arab that I showed in multiple events, that I rode that way. He was super light sided. So is my pony.
It's good to have them not so touchy, because if they lurch, you're naturally going to clamp onto them to stay on, but some are just more sensitive. You might want to consider not riding him in spurs, he may be too touchy and apprehensive about them to be calm. Unless there are times you do need them to get him through obstacles he doesn't like.
Many horses do have a good and bad side in the lope, and nothing gets them over that better than practice. One side feels nice, one side feels clumsy, and they often resist circling on the off side, they just don't feel as coordinated. It's one situation where a round pen might be beneficial, just because the hard barrier will keep them in the circle without them just bending at the neck and running out of the circle. He'll never get under himself doing that.