Hello, this is my car and I'm happy to answer any questions as best I can.
As far as the "Abarth" badge goes, it's a bit of an enigma. The car has been in my family since the 60's or 70's, and as far back as anyone can remember, the badge hes been in place. I can't say that it wasn't added by a misguided member of my family in it's early history, but it is a strange badge on its own. I've attached a little better picture (hard to photograph as it has chrome inlays), and I have never seen another Abarth badge like it. Even in a number of years of googling. It looks very similar to the early 60's badges that appeared on the racecars, with the more lifelike scorpion, however the legs don't match where they fall (relative to the red & yellow) on those examples. And of course "ABARTH &C." is absent from the arch of the crest. The badge is mounted via a threaded stem & nut on the back through a single hole in the fender.
I'm sure John can help here, but I don't know of a part, other than the battery, to be missing from the vehicle. Aside from the seat cushion on which the mice did a number , the only other exception may be drop-in side window screens, as there are a couple of fittings attached that appear to be for this purpose. However I don't know if they would have been original, or something improvised later.