Okay, let me attempt a couple of these -
1. How do you move your arms with the horse's movement at the canter without moving your shoulders or your body. Struggling!
Use your elbows. "Soft hands" is a misnomer in some respects; what you need is soft elbows. Your entire hand and forearm needs to move forward and back in as straght line towards the bit, by opening and closing your elbow.
If this problem occurs sitting the canter, but not in two or three point, it may be that you haven't mastered following seat in the canter, and your upper body is pumping to compensate.
2. My horse, Rusty, gets rather heavy at the canter still on the forehand. He likes to peanut roll his neck and lean on the bit. He doesn't usually do this at the trot, except he doesn't tuck his nose in at all then. I have him in a pelham and yet he still leans. He tracks up nicely in his gaits, so I know he is somewhat using his hind end. How do I get him to lift himself and not lean? What's the process? Is he shod behind? And with what? With a horse that resists carrying himself behind, one of the first things I try is to give a little more security behind and see if that helps. My fav choice is what's called a rim shoe or a polo shoe. Is the pelham a fat, rubber, mullen mouth? You might be better off with a very thin mouthpiece. The pelham is considered "more bit" because of the leverage action of the shank, and the curb chain, not necessarily because of the mouthpiece. Do you use the curb rein when you ride in the pelham? And when he is heavy, do you shorten the curb rein while asking him to carry himself? Next, better overall fitness. Then, lots and lots and lots of transitions, and hill work if that's a possiblity where you are. Post a video and I may be able to give you more specific advice.
3. Rusty has problems picking up his right lead. I'll ask, and he always picks up his left one. I'll bring him to a trot, and he picks it up correctly the second time only. Also, if we're doing simple lead changes, he'll pick it up then. If I bring him to a walk and start all over, Rusty responds to my leg asking for a canter as a shoulder-out. And I'm fairly sure I have it way back there. How do I work through this? Does he pick up both leads readily on the lunge line? Sorry to say that as you describe (difficulty with just the departure, not cantering in that direction)this is almost always a rider problem. There is most likely something in your position or in the way you're asking that's blocking the canter departure, something as simple as you ride a little heavy in your right stirrup/right seat bone or tend to look down going to the right. The other issue is insufficient respect for your leg aids/not forward off your let. More about that below.
4. Could anyone formulate a conditioning plan for me? I can usually ride 4-5 times a week, and I'm really just looking to get me and Rusty into better shape. I'd like to event him someday, but right now, I just want to condition and finesse him. I'm going to let someone else tackle this one in depth. I usually conditioned show hunters by hacking out 1 - 2X a week and doing lots of long trot sets and some hill work; don't know if that's available to you. There's a limit to how much you can do in a ring, in a hunter frame, to condition.
5. How do I get Rusty more responsive to leg pressure? We have trouble with going faster within a gait. I'll ask and he gives a small response (and I ask nice, then ask big), but then he'll immediately return to the last speed.
Ask, tell, demand. Carry a stick. The ask must be the tiniest aid you can give, a whisper. The tell should be the biggest aid you would ever want to give inside a show ring in front of a judge, and the demand should be tell, cubed, OMG, the world is coming to an end, and the sequence should be RAPID. If the horse hasn't responded correctly within 3 - 4 strides, dramatic things should happen! And reverting back to the previous speed should provoke an immediate correction. Most people make the mistake of tiny, incremental differences between the ask, tell, demand and that doesn't give the horse the proper incentive to listen for the ask. For leg aids, the sequence should be ask = tiny squeeze with calf, tell = light tap with lower leg or nudge with heel, demand = stick, three times hard behind the leg or until the horse shoots forward. The biggest mistake riders make is failing to be 100% consistent with this, or failing to follow up 100% of the time. You can't ignore lack of response to your leg aid because you're cooling out, there's someone else in the ring, you've lost your stirrup, you're working on your position etc., etc., etc., and you can't ever revert to squeeze/squeeze/squeeze/nudge/nudge/nudge/tap/tap/tap/kick/kick/kick nagging. I like to ask the horse to walk out as I'm leaving the mounting block and get in to them right then if I don't get a forward response, tends to set the correct expectation.