My horse was bred for speed and she loves to run. In the arena, she can get kind of hot at times and excited when anticipating. I also trail ride her and she's calm as can be. She has her occassional spook, but she doesn't try to bolt off or anything. I wouldn't run your horse on the pattern constantly, change of pace is always good and keeps them from getting bored.
As far as saddles go, I trail ride in my barrel saddle all the time since I don't have a trail saddle anymore. I'm comfortable riding in mine and feel more secure with the high cantle. Helps keep me in place when going up/down hills.