Because we know it, yes. We can see where the Sun is. You cannot see the Sun during the night, because it is on the other side of the planet.
(Notice the fallaciously circular reasoning - we know because we know)
Or, because you're assuming it. When you can't see the sun at night, you can't see it. So are you assuming it's whereabouts after that? You interpret your moon shadow one way, but I don't see you refuting the other interpretation I proposed. You're saying the sun is gone at night because it's on the other side of the supposed globe, but when you can't see it, how can you know where it is? Perhaps it is just further from you over a disk-like earth.
And again, how would you know? Are you assuming?
Because the left side is in the shadow, obviously.
Well how is that obvious? The earth covers the left side, therefore the
center is darkened? The moon gets lighter on a
gradient? The
center is darker because the earth covers the
left? The earth is not covering the
center of the moon in that photo, is it? Why is the center darker?
How could the earth create a softened shadow at the edges of the moon "shadow" with a single light source, unless the earth itself was a sort of fluffy planet with light gradually protruding through it? It appears obvious to me that the center in that photo is slightly shaded due to the sun creating a dusk/dawn effect on the moon, because it's on the right side of it, and that the shadow is from the moon itself, not the earth.
Then, why does the moon sometimes have a straight shadow down the middle? Does that mean the earth turned into a vertical disk? Or is it because the shadow is from the moon, and not the earth? Perhaps the sun is on one side of the moon, and the moon is in a triangular position relative to the earth and the sun.
Because it is not disappearing by moving away (light does not disappear with distance),
Of course light dissipates with distance. Go buy a cheap little flashlight and shine it on some distant trees at night. The effect of a local 'spotlight' sun is possible.
it goes "down" below the horizon,
I'd say it disappears at the horizon, but yes it appears to go down below. This may be due to it traveling further away from you. An airplane will do this from your perspective, as it flies overhead off into the horizon. When cars drive off, they appear to shrink, eventually into the horizon if the perspective is proper. They'll disappear bottom-up, too. And it is evident you can pull them fully back into frame with a telescope or camera if you zoom far enough. This seems to be how actual basic perspective works.
the same size, then you see just the half and then nothing. Again, watch a sunset.
You say the sun sets appearing the same size, but there is an entire study of atmospheric refraction to explain why the sun appears a different size at the horizon. It could be that the sun is to appear smaller in the horizon as a baseline since it's further away, but appears to distort in size due to atmospheric refraction.