I agree with the cold packs initially, on for 5 min,off for 5 min. I would also call your surgeon who did the implants and go in to see them to make sure nothing else is going on. Lots of swelling is not that common unless perhaps you had teeth removed,too?
To really align your teeth well and get the best possible result, i would do the braces first. The lateral incisors are tucked or lean in so the canines look more prominent.Bringing the laterals forward would help the look tremendously. the laterals do look small, so you may still want to do veneers on them when the braces are thru.
The first thing is to have the dentist adjust the bite so that it does not touch. If this doesn't thin the crown too much, it might be fine as is. Otherwise ,it would have to be redone but more tooth would have to be removed for appropriate thicknedd of the crown without being high
Thanks for the photos...it does appear that one veneer could solve the situation if it is matched well. Because the darker tooth sets back a little, the veneer can be made a little thicker to cover the darker gray color. It is difficult to match a single tooth...perhaps you could go to the lab for a custom shade to help insure that it matches closely.Good luck!
I would reconsider the orthodontic treatment..there is Invisalign and there are porcelain brackets which are 2 options that make having braces more esthetic. Invisalign , if you qualify, is also "less confining". I would not extract healthy teeth and place implants