So glad he's doing better!!! And now you know he's quick because he loves to go out. You're making him a happier horse :)
THe stopping thing is a common problem - pulling back on an untrained horse will freak them out. My mom's pony dumped my student the other day when she pulled back on him when he ran off with her. So I told her, you don't want to try to gain control back in a second. Because your horse is stronger than you and he will win. So, instead of trying to stop your horse quickly, take your time. Ride along with your horse, even if he's going fast, and try to take control back one step at a time. Ask for little victories, such as asking your horse to go to the left, then the right, essentially zig-zagging. This will get your horse back in a mind-set where he's listening to you, instead of freaking out. If your body pulls back and fights against him it will irritate or stress him out, but this way, if you're riding with him, you'll be more in touch with him and he'll be more likely to listen to you. Take as much time as you need to stop him. Once you manage to control his direction, pull him into a circle, like in the video, which will force him to slow down. Make the circle smaller and smaller until you feel safe to do a one-rein.
Here's a one-rein stop vid that will go over not just the one rein stop but how to teach your horse to stop from just your body language. I just made it yesterday. One Rein Stop - YouTube
If you want to instead teach your horse to stop when you pull back on both reins, teach him this at the walk, not trot/canter. Pull back, and then give right away when he stops (timing is key). Eventually he'll understand. I prefer the one-rein stop because even though it takes longer for the horse to understand, you no longer need reins to stop the horse - he'll stop from just your seat at the end of it. (if you want a vid of the result, let me know :) )
Here's another handy vid - it's about horses that REALLY misbehave under saddle, not really like yours, but it demonstrates the "yielding the hindquarters" exercise I mentioned (I still need to make a vid exclusively for this exercise) visually so that you can see the result. If you horse is goofing off, whether it's rearing or going way too fast, this will make them work harder and communicate to them that it's the wrong thing to do. Your horse doesn't seem mean; it really seems like a communication issue between the both of you.
Does he stand still for mounting?
Sorry for this novel!!! There are so many elements to this problem...Let me know how it all goes!