Dont be so hard on yourself.
Hey girl, don't be so hard on yourself. You are doing pretty well there, if you ask me. IMHO the first and last photo are the best. In the first I see a good leg position, and I wouldn't want me heel any lower for a lesson on the flat. You have a nice upright postion and good line from elbow to bit. The contact is weak and somewhat ineffectual, but maybe this is where the horse is in it's training; not ready for a stronger contact.
(one thing about the heel down thing. I was told to think more of "raising my toes" rather than jamming my heels down. Your instructor is right in that if you jam your heels down with more pressure than the weight of your whole leg itself, it will result in "pushing" your seat up and out of the saddle. Not good)
I was taught in English riding or dressage you NEVER bring the rein across the neck (mane). NEVER. You can use a kind of suppporting neck rein on the outside to turn the horse, but you still have to "suggest" a soft bend and an idea to the horse to turn to the insde by using some inside rein. you get the horse to "give" to the inside rein, to think that way and you support it by keep the outsid rein steady and firm against his neck.
I agree with another poster that a relatively firm and STEADY rein contact is more comfortable than one that is just loose enought that with each stride of the trot or walk it snaps tight, then loose, tight then loose. Very annoying to the horse.
Also, as they said, don't look so much to the inside. Look no further inside than your horse's inside ear.
Also, like she said, it takes a long , long time and many corrections to get the correct , vertical hand position. We ALL work on that one constantly.
I think you have the making's of a good, sypathetic seat. The horse seems happy enough. You are on the right track!