What I'm looking for is a plausible estimate. Would one freeze to death during night time for example, or would a temperature drop to "a slight chill" be believable.
Obvious answer: do you want people to freeze to death? Then fine: they freeze to death overnight.
Personally, I'd rather not. The long nights are obviously problematic in this regard, especially given a thin atmosphere at a high-albedo surface. I'm not sure what sort of vegetation could survive this kind of regular deep-freeze punishment, but I bet it woudl resemble lichen a lot more than it would resemble anything that might be described as a forest on earth.
Problem then: how do you increase insolation on the moon to ensure that everything doesn't freeze to death? Well, one way of fixing that is by arranging for a day length somewhat shorter than is currently available.
Gregory Benford's take on lunar terraforming involves hammering it with ice-rich asteroids to provide water and other volatiles whilst at the same time speeding up its rotation and maybe giving it a bit of a twist at the same time to arrange for some seasonal variation. The end result would be a 60 hour day and a deep, damp atmosphere that would be warm and cloudy. Benford compares it to a murky Florida, but doesn't specify which bits of Florida he was thinking of, and there's quite some variation. Most of florida is covered by a humid subtropical climate zone, which gets you winter temperatures around freezing and summer temperatures around 20-30 degrees C.
You might also consider orbital mirrors and sunshades to produce artificial days on the dark side and nights on the light side. These can be made bigger or smaller as required to ensure the optimum temperature, which is of course something that you would decide for yourself. This is perhaps slightly less destructive than hammering the moon with ice and rock for years at a time, and you might be able to make the mirrors and shades with the material you're processing to extract water and oxygen as part of the terraforming effort. Benford's Moon has a thicker, wetter atmosphere than yours, so solar heating in this way is probably essential. Heat would be lost quickly through the thin atmosphere, so your moon would be more like a cold desert... perhaps more like the Köppen BWk (cold desert) or BSk (cold semi-arid) classifications, with cooler summers (10-20 degrees C) and cold winters (-15-0 degrees C).
(FWIW, I think trying to terraform the moon is crazy, and paraterraforming with massive domes or even just building some nice big space stations in cislunar space would be much more sensible, so there you go).