Why don't dewdrops form on a cloudy night?
Clouds absorb thermal radiation and re-emit it, so that some of it comes back towards the Earth (Behind the Forecast: How clouds affect temperatures). During the day this warming effect is balanced by the cooling effect of blocking out the sunlight. They don't strictly speaking stop the formation of dew, but they make it less likely.

In the words of NASA's Climate for Kids pages, "It’s sort of like clouds are wrapping Earth in a big, warm blanket."

The 2nd question is about the effectiveness of greenhouse gases, and why we still get dew at night with all the extra greenhouse gases (GHGs) in the atmosphere. There are two parts to this. Firstly, the GHGs are not as effective as clouds when it comes to blocking the transmission of thermal radiation to space: they absorb in spectral bands related to the vibrational modes of the molecules in the gases, and there are gaps in these bands ("windows") which allow radiation to escape to space. 

Secondly, dew formation depends not only on the temperature at night, but also on the near-surface humidity. The increase is GHGs is, as we know, warming the Earth and this also leads to an increase in the amount of water vapour in the atmosphere, because warmer air can carry more water. To get dew formation we need a change in the temperature between night and day -- and GHGs don't contribute directly to this. 

There is an indirect effect which we see in nights getting warmer faster than days. There are several recent studies looking at the vapour pressure deficit (VPD) and showing an increase due to GHGs (e.g. Ficklin and Novick) -- VPD is a measure of the dryness of the air and an increases means less dew (though I have not found a reference that give direct evidence of reductions in the amount of dew). This drying of the night-time may also contribute to reduced harvests (Sadok and Jagadish).