I am currently in treatment for traditional braces with Dr John and could not be more happy! The assistants are wonderful and it is obvious they love what they do. I am so impressed with Dr John and his knowledge and the treatment I am getting that I wish I would have pursued this earlier. I would highly recommend McDonald Orthodontics to anyone!
From the photos, it appears that You have two separate but inter-related problems. 1) teeth that are not aligned (not "straight"). 2) A bad bite where your upper teeth are to far in front of your lowers and most of your teeth don't touch (an open bite). this is a very difficult combination to fix and it is not as simple as extraction vs non extraction. If it was done none extraction, your bite would get much worse and need jaw surgery to fix. Extractions with braces would make it possible to both straighten the teeth AND make the bite better (but probably not perfect) but avoid surgery. There is a 3rd option using TAD's but that would still require extractions. If you want to get this fixed, find someone (an orthodontist) really good who does not promise a simple solution and runs through some similar choices that I have just touched on.
What you see is probably not the lateral sticking out so much as it is the central incisor pushed in, or it could be a combination of both. Teeth only move when something changes. Every time you swallow the upper piercing (the barbell one) pushes back with lip pressure. Also, chewing on/playing with the lower piercing can also move teeth. Your gums are pretty inflamed and I see lots of plaque between your teeth which could account for the pain and certainly is the cause of the bleeding. (I see this type of tooth movement quite often in patients with similar piercing who come in for consultations with concern just like yours)
I agree with you that you smile is not nearly as good as it could be. There are multiple things that could be improved but I think your main concern is the shape of your teeth. That can easily be address with some reshaping of the sides of the teeth. Find a better orthodontist who sees what you see. It is up to you if you want to confront your previous orthodontist and ask for a refund or partial refund.
Premolar extraction cases are very difficult to do well with aligners. You have a combination of crowding and protrusion in both upper and lower and that is a tough combination for aligners. Wisdom tooth issue is a non starter, lower incisor is probably not enough space. If you don't create enough space in both arches, you will look "horsey" ie straight teeth that are to far forward. You have really nice long well formed teeth and show lots of teeth when you smile. This makes it fabulous when it is done right, but it also magnifies any thing that is not just right. Find a board certified orthodontist and ask for clear braces. You will not be disappointed.
It takes lots of factors coming together to make a nice smile, not just getting the teeth in line or "straight. I am just going to list some things you could look into, sorry there are so many, some people are just lucky that way. -Gums, yours are to thick and are actually covering up part of the tooth that should show. Google "altered passive eruption" and how to correct it. -Shape of teeth, yours are less than ideal. bonding or veneers would help -proportion: Length to width ratio of each individual tooth, as well as the size relationship to each other. Fixing the shape would also fix this. -Lateral incisor roots still to far back, so emergence profile is off. (this is impossible to correct with invisalign and is often undertreated in braces) -Incisors are to upright in the facial-lingual plane, also hard to fix with invisalign. -upper left lateral tipped, not enough space created to get it in place. Good news is all of this is fixable, it just takes a good orthodontist willing to do the work and address each of these issues. And it will take braces, but probably a year or less. Hope this helps, good luck.