Nice acquisition!
Welcome to the Board, and congrats on your new acquisition! I hope it turns out to be everything you expected.
The main components you want to check will be those engine parts that are specific to the 1962 build. The block should have a casting number ending in "068". Next up is the suffix code located on the front passenger side pad on that block. It should have a capital "T" for "Tonowanda" and then a series of numerics that hopefully will correspond to the sequence number on your VIN tag (that tag is the one located in the door jamb of the driver's side door).
The cowl tag on your firewall can tell you a bit about the vehicle, where it was made, what colorsfor the exterior and interior, and some accessory codes. The latter are harder to pinpoint, and may vary from plant to plant. Verne Frantz is the wizard in this area, you might contact him when you run into detailed questions. He lurks here as well
. He also maintains a "registry" of these X frame cars, and you can submit the details of what you know to him on a form so he can add that to his growing database. GM did not keep such records, so they need to be reconstructed from existing vehicles that are know or determined to be factory original.
That same cowl tag carries a body style, factory designation, and sequence number that can help you determine within a few days when your car was assembled. Any components on say, the engine, should have been made before that assembly date. Often the block and heads and other parts carry date codes that you can verify and compare to that "born on date".
It sounds like you are heading towards making this a "factory correct" or "period correct" vehicle, and if so you will need to document as many parts as possible. Things like the trans and such can be found (but probably not your original tranny) that might be considered "period correct" and you can put these back as replacements, if so desired.
There are other considerations, like ballast resistors, tachometers, and so forth that are clear indications of 1962 builds. Others can surely list these, as we have discussed this often (with some degree of agreement/disagreemnt!). I suspect someone can point you to previous discussions that have attempted to cover such topics.
Good luck with your sleuthing, I hope you turn up satisfactory results!
Cheers,
TomK