Ok I had the same problems with Lizzy. I got her this winter and she was a completely unbroken 19 year old pure arab. She still doesn't have brakes, they are better, no bolting anymore but not great.
How many different bits have you tried? It could be the bit was uncomfortable, hurt, too big in his mouth, there's many possibilities. I have not and will not ride my girl without a bit, too much risk of her bolting. She went through a rearing phase too, still does mini rears sometimes. But the easiest and quickest way to put an end to that is as soon as he pops up, while he is still in the motion of going up (this part is very important), take your hand and make a fist and hit hard between his ears. Sounds bad but it will hurt you more than him. They think they hit their head when they went up. Since he is a quarab he should learn very fast and that should work.
I would love to ride Lizzy in a sidepull, hack, bosal, eventually on the trails but at this point I don't think we will get there, she just has to much energy.
Oh and how do you check for wolf teeth when the farrier is there? He lets you put your hand all the way in his mouth? When was the last time his teeth were floated. I'm guessing there was a pain issue somewhere with the bit.