You are right, your toes need to point more forwards but don't worry that is a very common problem! The rocking motion at the canter is actually related, both of these problems stem from not having enough weight in the heel and lower leg.
Try doing lots of exercises without stirrups, it will help you to transfer some of this weight to your lower leg and develop a more balanced seat. This will help to lengthen your leg and position it along the girth (where it belongs) where your aids will be more effective. Just make sure you have nice posture when you do this by keeping your shoulders back and your head up.
The continuous kicking may be slightly annoying for your horse but he really isn't responding much by stretching out and moving partly because at the moment you are kind of 'nudging' him with your calf. Instead you need to use firm heel aids, just behind the girth and if he doesn't respond give him a tap with the whip as the first post mentioned - he should listen to you! Incorporate some trotting poles also to encourage him to stretch out at the trot.
Your reins are also a little loose for my liking, you should have constant, even
contact with him so that you can 'push' him onto the bit more. His movement will improve vastly if you can get him to be a little lighter and more forward, he looks like he is on a leisurely Sunday ride at the moment!
Your horse is a cutie though - what a beautiful tail! I always notice a nice tail as most of my horses had wimpy thin ones ha ha
All horses deserve, at least once in their lives, to be loved by a little girl.