E: I'm still a bit stumped as to the exact source of this case, unfortunately. Regarding the movement, Andy already called it: It's a Sea-Gull ST-19 aka 'Venus' aka 'Lemania'. This is the go-to movement for nearly all hand-wind chronograph duties in the rep world, and a reliable one, too.
As for the crown, 703 would be best on the 6263. It's a bit hard to see the current crown in your pictures, but I'll bet you'll see a real improvement with the gen. If not, you'll at least feel the superiority of the gen crown.
Dimensions for genuine-spec pushers (e.g. Ofrei, CousinsUK, or Rolex itself): 0.25mm diameter, 0.2 thread pitch.