The rearing, if it was a one off let it pass but be aware. Next time he does it, run him full pelt backwards then backwards some more. Teach the brute that rearing is **** uncomfortable. If he gets away with it he may begin to strike and that is extremely dangerous!
As for the problems under saddle. He's a young tb with a new owner/riding. he is going to test the boundaries, that's just what they do.
Keep him forward and thinking. Lots of circles, transitions, changes of rein, serpentines, leg yield etc. etc. Don't go more than a few strides without changing something about how he is going. You need to keep him thinking so that he's more focussed on the next thing you're going to ask than how to get out of the work.
When he bucks, really get up him, force him to go forward and really drive him up to the bridle. If he is the type of horse that you can feel a buck coming on, pull his head around and spin him on a small circle, then ride out again.
The crow hopping is a matter of not having him forward, they can't do it if they are travelling so when he goes to start crow hopping, get up him and get him choofing along so that he cant get the momentum to crow hop.
I don't take rearers. Not into that type of dirt. He can't rear if he's going forward, so keep him active. If he stops and jacks trying to go up, again, spin him and kick him out. It's far better to avoid the rear than get up into a vertical rear where he can flip over and badly injure you.
Horses that get into the habit of rearing, I've heard that cracking an egg over their head at the height of the rear will make them think they've cracked their skull and they won't try it again. I've never had to go to that extreme to 'fix' a rearer, but a friend did so on her ottb and he has never reared since.