...Question 1: Ok, there is a difficulty with reins - the horse has a tendency to go down too much while cantering thus sometimes he stumble. The trainer told to me to bring his head back and show him that he has to concentrate and see everything. It might look as if I was pulling him but I really tried to have flexible hands. So what do you recommend? Loose the reins, make them longer?...
When I first started cantering on my mare, she'd go way heavy on the front end and put her head down so low that I was afraid we were going to go arse over tincups. I don't have any pictures, but she'd put her nose to within inches of the ground while doing a canter that was just under a gallop. Add in that my arena is too small for her to gallop, and it was an ugly feeling.
So the first thing I did was simply pull her nose up. I'd start the canter with a loose rein (I normally use an Aussie-style saddle and western-style slack in the reins) and then if she went head low, I'd brute force her head out of the ground. Oddly enough, I didn't have to brace against the poleys. It was mostly a matter of 'toes up' (I hate heels down), keep most of my weight in my thighs, force my shoulders back and pull.
Once I got her nose off the ground, I found I was encouraging her to put too much weight on her front by - you may have guessed it - leaning too far forward. FWIW, the mental image I used (use) is to keep my weight in my thighs and tilt my shoulder back behind my thighs, without my rump settling into the saddle. No pictures of me doing it, but my GUESS is that in reality I am still SLIGHTLY tilted forward & it only seems otherwise in comparison to what I did before.
Another thing that worked well for me was to use a restraining rein. Although I prefer slack in the reins and even neck reining, when Mia gets wound up I go back to two hands. As the right shoulder goes forward, I don't pull back, but I lag a little in moving my right hand forward. It isn't so much a 'slow down' as it is a 'don't reach so far with each stride'. We've practiced it a lot at the walk & trot, so she picks it up fast at the canter. In 6-8 strides, she'll keep driving with her rear but take shorter steps with her front, and we end up with her shifting her weight to the rear. This usually raises her head more, but she's Arabian and I don't do any showing and it works for us.
With both my Arabian mare and my mostly Arabian Appy gelding, trying to hold them back using both reins seems to make them want to extend forward MORE, not less. It becomes a tug of war that I either lose, or that forces me to bring them down to a walk.
Shifting MY weight back - not sinking into the saddle but just shifting my balance - and using a restraining rein results in them shifting their balance and loosening up.
I've had perhaps 20 lessons in my life and am pretty unorthodox in my style, so take all this with a huge FWIW.
Quick note on heels down...my heels don't go down very far. However, when my heels come up, it usually means I'm gripping with my knees instead of allowing the weight to slide past my knees and into my heels. I have tight hips, so I often need to make a conscious effort to push my knees apart. FWIW.