Hi,
Yes you can make a 650cc Giardiniera- Yes you use 126 pistons and barrels and I have built one, it is a lot more involved than simply bolting a 126 engine into a 500. You are very lucky that you have a 'F;' series wagon which has stronger clutch, axles and driveshaft couplings which is ideal for converting to 650.
From memory gearing is unchanged from the sedan- I have a Fiat 126 transaxle in mine which has a taller final drive and also a 5 speed conversion which is useful if you are planning to do motorway driving.
Almost everything is different in the Giardiniera engine except Pistons, Valves, Crank and bearings so you will need a complete donor Giardiniera engine if you want to retain the original.
Personally I wouldn't bother as once you have driven a 650 you would never go back to the 500 and there are no visible external modifications once it is assembled- I have two original 500 engines out of my sedans, one has been out of the car for 23 years, the other 10 and at no point have I been tempted to reinstall them!
As has been advised Torsten at il motore in Hamburg is the best person to talk to- best to call him as he isn't great on responding to emails and his English is good (he even understands Australian!)
Good luck!