the horse has a lot of drive from behind and would probably be good in lower level dressage. I agree with Sky, he is behind the vertical and in a curb bit I would think there's a fair amount of pressure on his mouth but I am not holding the reins, so don't know the truth of that one.
Him jumping that gate is really scary. the top bar is literally brushing against his rib cage. I keep imagiining him flipping, cartwheeling, getting his front legs in the slats and snapped like twigs.
My lease horse was doing this, too. Jumping the gate from a stand still (he's an appy, too, btw) when the gate was allowed to hang less than upright. I hope you will fix that gate and string some hotwire to discrouage him from trying again.
But, back to your question . . . . I would look into dressage. or jumping. He does look a bit forward for any kind of Western showing . pretty horse!
ETA ok, it looks like you are in a snaffle. I just can't see quite clearly. AND not sure if the top rail of the gate is missing where the horse is coming over or not. again, picture clarity. horse sure can tuck his knees nice.