For wave frequencies well below the resonance frequency of the scattering particle, the amount of scattering is inversely proportional to the fourth power of the wavelength.
This wavelength dependence of the Rayleigh scattering (~1/ Lambda^4) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths.
This results in the indirect blue light coming from all regions of the sky.
Here, Rayleigh scattering primarily occurs through sunlight's interaction with randomly located air molecules. It is this scattered light that gives the surrounding sky its brightness and its colour.
In addition, the oxygen in the Earth's atmosphere absorbs wavelengths at the edge of the ultra-violet region of the spectrum. The resulting color, which appears like a pale blue, actually is a mixture of all the scattered colours, mainly blue and green.
Conversely, glancing toward the sun, the colours that were not scattered away — the longer wavelengths such as red and yellow light — are directly visible, giving the sun itself a slightly yellowish hue.
Some of the scatterings can also be from sulfate particles. For years after large Plinian eruptions, the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime.[11]
The scattering of red and green colours do take place but the blue masks them off.
Coming to the II nd part of your question-
The fact that the sky is not completely dark at night, even in the absence of moonlight and city lights, can be easily observed since if the sky were absolutely dark, one would not be able to see the silhouette of an object against the sky.
In the twilight, the period of time between sunset and sunrise, the situation is more complicated and a further differentiation is required. Twilight is divided into three segments according to how far the sun is below the horizon in segments of 6°.
After sunset, the civil twilight sets in, and ends when the sun drops more than 6° below the horizon. This is followed by the nautical twilight when the sun reaches heights of -6° and -12°, after which comes the astronomical twilight defined as the period from -12° to -18°. When the sun drops more than 18° below the horizon the sky generally attains its minimum brightness.
Several sources can be identified as the source of the intrinsic brightness of the sky, namely airglow, indirect scattering of sunlight, scattering of starlight, and artificial light pollution.
There are interesting images of the star glow(sky being lighted) when our own nebula is observed.
References-
https://en.wikipedia.org/wiki/Rayleigh_scattering
https://en.wikipedia.org/wiki/Night_sky
