To move teeth you need invisalign or something similar. Your overall bite can be affected by the slightest shift. Some dentist can fabricate a special aligner that can shift only the front tooth slightly without going through an entire invisalign case but not every dentist feels comfortable doing this. Another option is to do enamelplasty which re-shapes the tooth a bit and gives the illusion the tooth is aligned more symetrically. This involves removing millimeter(s) of tooth structure but is safe when done correctly. You may need to reshape both front teeth, possibly add bonding to achieve optimal look. This should cost about $500 or less.
It is a good thing that you had the scaling done. Your gums are the foundation on your mouth and periodontal disease leads to bone/tooth loss. Keep it in check and go back for cleanings every 3-4 months. Also ask your dentist about bonding the lower front teeth to close the space a bit and smooth out the sharpness. Bonding is applying tooth colored filling material to the teeth. It is used in fillings and fixing chipped teeth. If there is significant gum recession in the area, a gum graft is needed to replace the gum tissue. It is painful and doesnt alway work. Go back to your dentist and discuss the situation with them.
A combination of Orthodontics and veneers will help. Also subtle lip filler can change the way your lip looks when you smile. You can achieve what you are looking for but it will take time and money. Consult an orthodontist as well as a cosmetic dentist.
Your dentist recommended Zoom because the new veneer need a shade/color. If you are happy with your current shade on the natural teeth, then tell him or her that. If you feel you want to lighten your natural teeth and match those with new veneers, then do the whitening/zoom. The veneers will not change color but your natural teeth will only get darker...bleach trays for 'touch-ups' are also useful when you have veneers, next to natural teeth. A1 is light but B1 is the lightest shade...probably too white. Have an in-depth conversation with the dentist about this. I would recommend doing the bleaching before the veneers but if you do not see yourself maintaining it with trays at home and you are happy with your current shade, then skip it.
Your situation is not permanent and your gums will heal. Try warm salt water rinses and vitamin E to help it heal faster. The dental assistant was probably inexperienced in doing the whitening for this type of situation to happen. I would go back and have the dentist take a look and explain to him or her how much your soft tissue was not properly blocked/protected. There is a very basic step when performing a in-office teeth bleaching and that is to block / protect the gums from the bleach. Your mouth is also to remain open with a bite block, not allowing for the bleach to touch ANY soft tissue in the mouth. Some dentists will apply a material to protect the gums or a rubber dam. It seems like a very sloppy in-office bleaching job (by someone very inexperienced) to me.