Why does the third quarter moon rise at midnight and can be seen anywhere on Earth at midnight?

Answer 1

The answer is in the geometry of the earth-moon-sun arrangement as it changes and repeats each month. It's easiest to show in a picture, but since I can't draw one, I'll have to point you at one.

http://www.astro.umd.edu/education/astro/moon/phases.html

During the 29 day lunar month, the earth spins about 29 times. The third quarter phase isn't really instantaneous, so it appears to be third quarter for about a day or so out of those 29 days. Since the earth and moon have to be lined up the same way for the moon to appear as third quarter, and the moon is always in the same relative position in the sky during third quarter, it always rises at the same time in relation to the sun as it appears to you the earthbound observer.

If if rises at any other time, it won't be lined up to look like a third quarter moon since the sun will be shining on a different side of it and you'll see a different illuminated fraction.

Answer 2

3rd quarter is defined as the Moon being 3/4 of the way around its orbit starting from new. At such a location, it would rise 3/4 of a day after the Sun, or 1/4 of a day after sunset. So it is just geometry and definition.

And it can only be approximately seen from everywhere on Earth. It is only exactly at 3rd quarter at one instant in time, at which time it can only be seen from half the Earth. Of course over the next half day the Earth rotates around so that the other half can see it, and it is still close to 3rd quarter at that time.