Okay, here;s my 2 cts. It looks like a unmolested 66 gt, so you need to decide whether to keep it original where it has the most value, or to make changes and/or mods, if you do, then I would save every part you take off, so if you do decide to sell it later the new owner might want to make it original again. Just looking at the pics, your GT emblem on the back is your gas cap and I don't see the filler neck for the tank, some when working on the car, turn the filler neck inside the tank and put the gas cap back on, until they are done with whatever repairs they are doing in the rear of the car. It also looks like you need to reupholster at least the front bucket seats. It has fog lights and a/c both good options, the a/c may only need the o-rings replaced and freoned up unless the front seal is bad in the compressor, those compressors are as rugged as they get. Is that just dirt on the paint or is it worn off, if it is just dirt, Then I would wash it and use a good paste wax. Any rust needs to be addressed, either removed and patched or at least sealed until a later date to fix it right. And you can pat yourself on the back, it looks great for a 49 year old car.
I decoded your door tag using my books and I come up with some discrepancies, you should compare the serial number stamped onto the fender apron by the shock tower on the drivers side to verify the door plate is the one for this vehicle, if it had been wrecked before on a drivers door they would just change the door and not transfer the plate, I will list the decode and note the ones I found:
Body- 63A- 2 door Fastback standard bucket seats
Color-R- Ivy green or Dark Green Metallic (my books differ)
Trim-26-Interior Black
Date- 03J- September 03 (1966)
DSO-65-Oklahoma City
Axle-A-3.00:1 axle Equa-Lock Differential (this is an option, I would look at the tag on the differential to hopefully verify it, the only other way, is to drive the car like on a wet surface, and spin the rear tires, the differential will lock up so both wheels are together, to unlock, back the car up several feet and the rear axle will unlock the tires, normally this axle acts just like a normal axle otherwise. These are great rear axles)
Trans-5- 4 speed manual (It looks like you have an automatic C4 which should be a code of 6)
Serial number
6- 1966
R-San Jose Assembly plant
09-Fastback
A-289 4v premium fuel
104583-serial number (mustangs, falcons, and fairlanes combined in this)
Good Luck. One real upgrade, I recommend is to change the brake master cylinder to a dual-chamber design like on the 1967 mustang, as with the single chamber like you have, is if you develop a brake leak, and loose the fluid (you won't know till it is gone) you will have no brakes, and will have to use the parking brake to stop the car, been there and done that many times, I have since upgraded my 65 to a 67 dual chamber. If you are going to keep the car original and have it judged in shows, then I would leave the MC alone.