1. Aside from check your tack and check for pain issues which could explain the rearing but not the not stopping issue.
2. Has anything changed in his life? New tack, new feed, new home, new rider, new trainer, turnout time change, anything at all over the past year. I ask because horse are animals of habit. They don't always do well with change.
3. When you say, "He does have some quirks but they are not serious." I have seen, "some quirks" meaning doesn't like the garden hoes or bug spray to things like; tries to bite when I saddle him, to tries to kick when I feed him. So when you say "some quirks" I don't know what that means and how it might or might not have any effect on him rearing.
4. Rearing is normally a lack of forward motion issue, but it sounds like this is not the problem seeing as you are having trouble getting your horse to stop. How do you ask your horse to slow down or stop? If all you are doing is pulling back your horse is not getting the message that pulling means stop so his brain is saying go, go, go but you are pulling back keeping him from go, go, go so what you get is not go, go, go you get up, up, up. I would work on slowing and stopping him without pulling back. Some people are big fans of "one rein stop" that would be one idea, circling, asking for the stop at the same time asking him to step his hindquarters over, pulling his head around hard (this would be a last resort).
Remember horses are pulling animals they are most powerful when they are lined up ears, withers, tail when they are in a strait line they have all the power, break that line and you are robbing your horse of power. Turn his head, or move his hips over are great ways to break that line. I like the hips myself I find it works really well. Other then that I don't know answering these questions would help us understand you horse better.