What procedure would best correct my horribly recessed chin/jaw? (Photo)

If you have a bad bite and wish that to be corrected as well, than braces followed by lower jaw advancement would be the correct treatment. This may also help your TMJ clicking and popping. If your bite is not a concern then a sliding bony genioplasty or chin implant with submental/neck liposuction would be the other options. Given your shorter neck, a sliding genioplasty would create a better neck angle as it lengthens the jaw bone and pulls the anterior tongue/neck muscles forward.

A sliding genioplasty is performed as an inpatient procedure by an oral surgeon with at least a one night hospital stay. A chin implant can be placed for cosmetic purposes and is performed through a small sub mental insertion under local anesthesia as an outpatient surgical procedure. Orthodontic braces can help with straightening crooked teeth. From more information and many examples,  please see the link and the video below

your lower jaw is underdeveloped and the 'bite' appears to correlate with that issue. The more ideal approach is to undergo orthodontics and 'lower jaw advancement with a possible genioplasty' to correct your concerns. This is a common situation and you need not worry about the results being quite predictable.