The earth's shadow can sometimes create a crescent-like appearance when the moon passes through it, but the earth would have to be between the sun and the moon, (to cast a shadow). That could not be the case in the composite shown, regardless of the size inaccuracies.

The normal phases of the moon are very different and  caused by our viewing angle. In that case there is no "shadow" blocking sunlight to the moon. Half of it is always illuminated. It's our relative position that allows us to see more or less of the lit half of the sphere.

It might help to visualize that viewed from the moon, the earth goes through phases too, only they are the reverse of how the moon looks to us. For instance, when we see a tiny sliver of the moon illuminated, a nearly "full" earth is visible from the moon.

In answer to your second question, the sun and that thin a sliver of moon would never be visible together on earth. The diffusion of sunlight through the atmosphere makes it impossible. You can observe that those thin crescents are only visible after sunset. And the thinner the crescent, the harder it is to catch between sunset and when the moon sets.

One other fun fact. Have you ever noticed that you sometimes see a crescent moon and the rest of the disc seems lit up by a very faint light?  Remember that a crescent moon to us means that the earth is almost full (the reverse phase) as seen from the moon.  So that faint illumination is sunlight bounced off the almost "full" earth, and bounced off the dark side of the moon, back to us! I always thought that was cool.