I personally like to have a frenectomy done after orthodontics (whether you use braces or Invisalign). The reason is that when you do any type of surgery, you are going to have some scaring occur. I would rather have that scaring form around teeth that are closed versus open. That way, it would have a tendency to keep the teeth together (although another form of retention would be needed too). My two cents...
It appears from your photo that the extraction of teeth would not be necessary to give you the smile you deserve. Of course, an extensive exam would be needed to fully diagnose and treatment plan. With the Damon System Braces that we use....the % of extraction of the teeth has gone from 30-35% of all patients to less than 1% of our patients. I would encourage you to go to the Damon Locator website and find a doc in your area.
It appears that a removable retainer could possibly be used to close the front space and to move the one tooth back some. This would be the most conservative treatment and the most economical. However, I would advise to do a full, extensive exam with your orthodontist to see if this limited type of treatment would be indicated. As with any type of limited types of treatment, you usually get a limited result....but sometimes the limited result will address the patients chief concern.
I appears to me that you have a skeletal open bite which is an underlying jaw problem. Of course a full orthodontic examination would be needed to fully assess. Although Invisalign is great a closing anterior open bites in dental malocclusions, skeletal closure is a surgical correction that is sometimes indicated. Again, full orthodontic records and a detailed clinical exam would be indicated.
A 9mm overjet is pretty significant and should be addressed as early as possible. It sounds like a Class II corrector is needed to be able to correct your malocclusion if you are still in a growth spurt. Otherwise, a surgical option may be the only other options you have to correct. I would imagine that correcting this overjet would have a profound change on your occlusion and dental health as well as a profile change that may be warranted. I would seek an orthodontic consultation as soon as possible to evaluate your options!