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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.
It is unfortunate that you had four teeth amputated and no have these issues that could have been avoided by early expansion. Consider the use of the DNA Appliance it uses very light forces and there is virtually no inflamation which causes pain and resorbtion and/or bone loss. Shortened roots...
Orthodontic treatment will definetely close the gaps and correct the bite.Post orthodontics the crown will be in better shape and the bite will be uniform.
I don't recommend orthodontic treatment until you are ready to stop the thumb habit because it will likely be a waste of money. Continuing the habit will undue all of the work with braces or Invisalign, and a retainer will not likely keep the teeth in a good position.