Kind of depends on which mark, but yes, the book says to overfill it somewhat and the level will drop from there. The excess oil runs down between the shaft & dome, to lubricate the sliding fit. As long as you can feel some resistance when inserting the damper, there is enough oil for proper engine operation.
The later ZS carbs have an O-ring that can leak oil; but on the TR3 SUs it is one piece and so very unlikely to leak.
The dampers basically give a richer mixture for acceleration, because that gives more power. It also helps accelerate the fuel through the jet (otherwise the mixture would go lean just when you want it to go rich). Very similar to an accelerator pump on a conventional carb. By keeping the piston from rising as fast as it would otherwise, the air velocity through the venturi is increased, which pulls more fuel through the jet.
Likely it varies depending on your engine's condition, state of tune, etc.; but what I found was that 20 weight motor oil worked best for me. ATF produced a definite "lean bog", while heavier oil made it harder to apply power smoothly without adding anything to performance. The racers use ATF or even nothing at all, but they also have their jets modified to run full rich all the time. For a street-driven car, it's better to have the cruise mixture a bit on the lean side (for less fuel consumption) and richen it only for acceleration (for better power). This is the stock setup. The owner's manual says "current engine oil", which would mean 40 weight for temps consistently over 20C, and 30 weight for temps between 0 and 20 (etc).
When the carbs are cold (which doesn't necessarily correspond to colder weather, since the carbs are warmed by engine heat), the effect is increased giving a little extra mixture boost for acceleration, kind of like a bit extra choke. But that is appropriate since the fuel also won't vaporize as well, so the engine actually runs better with a little extra fuel.