Inside leg outside rein, also, a good warmup. If your warmup thoroughly relaxes and loosens the horse, they should also relax when the harder training starts and with that keep the head in a place in which the back also relaxes (so more down).
to slow your horse when they bolt is to pull them into circles and do smaller circles each one.
"Turn your key" take your reins and as you pull a little bit turn them like you would turn a key to unlock the door.
I also would recommend keeping your hands low. When you do this it forces their head down. Also, do continuous half halts and release the pressure when their head is down.
When transitioning into the Trot and canter tweak the reins to keep their head down during the head down. When in the hair don't force anything keep your hands supple and slightly tweak to get the horse on the but and supple. Also there are many different excerdises when ridding you can do that will help with this.