When
One synodic month is 29.53 days — New Moon to New Moon.
Atmosphere
Rayleigh's law sets n = 4. Drag it to 0 to switch wavelength dependence off and watch the sky turn grey.
Dust and droplets are far larger than a light wave, so they scatter every colour almost equally — that's the white glare around the Sun.
The Moon now
The Sun now
Scattered light by wavelength
Air molecules scatter short waves far harder than long ones. Blue light leaves the beam in every direction and reaches you from the whole dome — so the dome looks blue.
Near the horizon sunlight crosses tens of times more air. Blue is scattered out entirely before it arrives, leaving the red end: the same law paints both the blue noon and the red sunset.
The Moon makes no light of its own. Half of it is always lit; the phase is simply how much of that lit half faces you, set by the angle between Sun and Moon in your sky.