For chrome and paint try Velociao in Berlin. Only place I would trust with electrolytic chrome removal. Pricing is sane, they do a lot of this, and a lot of Cinellis. All staff speak English.

Of course if you did chrome and paint that would call for new or newish looking parts as well. You could just wax that chrome. Do you like orange? Getting rid of the orange would decide it for me.

Probably not any original parts except headset, cranks, stem and maybe handlebars. Headset, cranks and stem are all rare and definitive for this bike. Keep the headset even if you have to have the races re-lapped. And if it still runs smooth be happy. Crank takes either 116bcd or 151 chainrings, either very available. Alloy T.A. Criterium rings are easy to get and would not be wrong here.