Ironically, the dark side of the moon is lit a bit more than the light side. Dark side is gonna be almost exactly half of the time, light side is a bit less because it's the side facing the sun during lunar eclipses.

Same argument for the dark side of the moon getting more light than the earth because of solar eclipses (if we're comparing % light coverage, not total photons).

Depends on the definition. Ignoring the terrain, each point gets slightly more illumination than 50% of the time because of the Sun's non-zero size.

How often is the moon completely invisible in our night sky? We reflect to the moon the same way the moon reflects light to earth, but that is the visible side, on the other side you have all the stars, gravitational lensing, other planets reflecting to that side of the moon, and even the small amount of luminescence all matter emits. But for easily visible on the dark side I'm pretty sure if there isn't a small amount of the "dark side of the moon" lit up by the sun that it will appear like a lunar eclipse on earth minus the effect of our atmosphere, or it might be more of a solar eclipse situation and it would just be black with a very bright halo or outline.