The dropping idle and stall is probably the rich running - mixture suits the cold engine, but is too rich when the engine is warm. If adjusting idling mixture does not fix it, then it is probably a carburettor issue. I don't remember what carbie you have, but with the Zenith this is usually the O-ring not sealing perfectly. There is a slight possibility that it is something else, such as a heat sensitive part of the ignition system, or that the engine is actually overheating.
The base of the carburettor will normally get cold - evaporating petrol takes heat. Nothing to worry about until you get ice build up internally - never heard of that happening in this engine. That is why the exhaust manifold is attached to the inlet manifold. Early exhaust manifolds had a thermostat controlled deflector that adjusted the amount of heat, but later ones dropped this. Probably reduces maximum power slightly.

