As to oil to use in the damper, it might help to understand the reason it’s there.
CD (constant depression) carburetors, like S.U. and Zenith-Stromberg carbs, rely upon a consistent vacuum pressure over the fuel jet to properly meter the fuel over the full range of acceleration.
In non-CD carbs there is usually an accelerator pump that squirts a shot of gas into the intake stream when you hit the throttle. This is because to get a quick throttle response, the engine needs a momentary extra-rich condition, or else you’ll have a momentary flat spot on acceleration.
In CD carburetors they compensated for the lack of an accelerator pump by using an oil-filled ‘shock absorber’ of sorts - and that’s the damper you fill up.
When you open the throttle quickly, this damper temporarily delays the rise of the air valve (the bit with the fuel metering needle), and thereby briefly increases the vacuum over the fuel jet. This means that the intake stream gets a briefly richer mixture, in effect creating an ersatz accelerator pump.
A heavier oil delays that movement longer, giving a longer time of richness, or essentially a bigger accelerator pump ‘squirt’. Conversely, a lighter oil gives a smaller squirt.
Also, one has to take temperature into effect. As noted in the previous posts, a single-weight oil heats up it gets thinner, whereas a multi-weight oil tends to hold a more stable viscosity over a broader range of temperatures.
So, in choosing the correct oil, one should use an oil that gives you enough delay such that you don’t feel a flat-spot on accelerating, and it should do so both when the engine is cold and hot.
I use 10W-30 engine oil in my carbs, as it gives me nice acceleration throughout the normal operating temperature of my motors.