Sounds like you have experienced the far extremes of horse training. Somewhere is the middle is more healthy for the horse and rider.
To have a horse/pony roll with a rider on it's back is unexceptable, and dangerous. It is a lack of repect by the animal.
Too beat a horse to get it's attention does not teach the horse repect, but fear. It is considered the old cowboy way. It is effective to point, but in the long run not healthy for the horse or rider. Usually it is a lack of patience by the trainer.
The correct way in my humble opinion would be to build a foundation of repect on the ground, being firm, but not cruel. Then building on that foundation.
There is a time and place to get the attention of a horse, but it should not be done in anger. But rather pressure the horse to find the correct answer to it's behavior. EXample, This pressure may start off by spanking the horses hid quarters with a soft cotton rope to get it to dis-engage it's hind quarters, this physical action is more startling then scary, and soon enough you will only have to lift the lead line towards the hind quarters and the horse will dis-engage. Remembering that if you want the horse to find the right answer you will use only enough pressure, physcially or physcologically as necessary. The less pressure you use the better for the horse.
On the other end, to allow or assist an animal to make decisions that are dangerous to a rider, such as rolling with a rider on it's back is just a bad judgement. The use of a crop to tap or spank the horse once to get it's attention back on the rider would be more appropriate. Not to let it roll, causing the rider to bail off the horse.
As for the horse or pony not holding it's head correctly.
If the horse is not holding it's head in the correct position then there is a reason. Some horses are more forgiving then others with novice riders. With the head of a horse, that horse will always try to find release from pressure. We want the horse to have both lateral and vertical flextion, finding the release by tucking it's head slightly. The trainer instead of beating the horse may have been better served by having you turn the head of the horse from left to right until it became softer to the bit. Then continuing your lesson.
Good Luck and be safe