Definately what Puck said so I'm not going to repeat that as it is excellent advice! Only thing is he can still just run onto his forehand through these transitions, so make sure your keep you back and leg on him. If he stiffens, turn him onto a 15 m circle (10 is better but he's so stiff and on the forehand at the moment he would struggle with a tight turn) and kick him off your inside leg, keeping your outside rein back on the saddle cloth. Then ask him back out onto the track with a few strides of leg yield to get there.
As for the catch 22 situation with muscling, not so. HILL WORK girl :P It is fantastic to build the muscles he'll need to be able to come off the forehand. A nice active working trot up hills for 10 mins or so each day (depending on his fitness level) will really help to build those muscles.
Back in the arena, transitions transitions transitions!! Trot-canter-trot transitions are the best you can do for putting them back over their hindquarters so plenty of those, not letting him run into them. Use your back to hold him up in the transition.
Try doing 12 strides of trot, 12 strides of canter, 12 strides of trot.... it will keep him thinking and won't give him much of a chance to fall on his face ;)
Lots of changes of rein will help improve his balance, start with 3 loop serpentines and work up to 5 loop, then 6 loop once he is balanced enough.
Put him on a 20m circle and work him into your outside rein. So lots of inside leg, keep your outside rein on the saddle cloth (DONT give it away!!) and lift your inside rein up and towards the inside of the circle. This isn't the most conventional way of doing things but I have found it works well with horses that are as stiff and strung out as he is. The inside rein is there to ask his neck and jaw around to the inside, and with the inside leg pushing him out, he will give to the inside rein. When this happens, give immediately (not throw the reins away, just take the pressure off) and let him trot for a few strides. Then pressure back on and ask again.
Leg yield is also a great excercise for starting a contact from inside leg to outside rein. get your instructor to teach you how to do it. Once leg yield is established shoulde rin can be started, but he will need to have at least a little understanding on softening his jaw and stepping under himself before he can achieve shoulder in.
Turn on the forehand is great as well for submission and learning to step under themselves. If the horse has learn this and starts to resist and hollow during other work, I will immediately ask for a slight turn on the forehand no matter what the pace I am travelling at. It forces the inside hind to step under and take the weight, and the jaw will soften as a result.
As for your position, you are balanced on your toes and you're 'driving a truck'. Sit back, sink your weight down through your seat and down to your heels, roll your shoulders back and down, and imagine that your elbows are the heaviest part of your arm, keeping your hands and wrists 'light weight'.