What should I work on fist when starting a horse on dressage?
In my opinion connection is the basis for everything! If your horse is ignoring the bit then you can't ask for anything! Neither you or your horse should pull on each other. Just be able to feel the horse on the bit and it be able to feel you as well. Get your horse well rounded and moving more off of body movement. You should not be steering with your hands. I hope this helps a bit :)
Do we have more link with our own horse or with a school horse, and why?
Your own horse. Though I will say I'm currently riding my trainers one horse who she will occasionally use for lessons. And we do have a bond. However I am his main rider and he is only ridden by anyone else maybe once every month or too. I also spend time with him outside if lessons and riding. I have the opportunity to do much more with him than an ordinary "lesson horse"
Is it a sign of pain when a pony is trying to bite whenever we try to do the middle or front straps of the rug? I think her rug is too tight as well.
I would say either don't use the rug or get a bigger one. If it's tight enough that you think it's to tight it most likely is. Your horse should have room to move! :)
How can I build up my horse's top line as well as build up some more muscle in myself?
For the horse you can try lunging or riding with side reigns. I wouldn't recommend always doing this but one or two times a week will help build the top line. And for you just basic stretches or a basic work or routine. Crunches, push-ups, planks, holding your legs off the ground with your back and but flat to the ground, ect.
How to build a horse's confidence over the jumps?
Just thought of something. If your horse is running around the jump do you ride with a whip? Keep the whip on the side he is running out on and LIGHTLY tap him with it on that shoulder coming up to the jump. I'm not saying abuse your horse and I'm not saying tap so lightly they can't even feel it. There is a fine line with whips as so many people think they are automatically just abuse :/ do this even if he doesn't run out. Once the running out stop start to stop doing the tabs to show him you trust him and don't let it turn into a robotic movement.