Why should our time keeping on Earth have anything to do with the moon?

Because the moon is a big bright ball in the sky which regularly moves through predictable, easily observable phases, making it an obvious basis for a system of measuring time.

The complexities, irregularities, and imprecisions of our calendars are due to things like

* the Earth's rotational period ("day")

* the Moon's orbital period ("month")

* the Earth's orbital period ("year")

not being integer multiples of each other at all! Despite that, we naturally want to keep time in terms of each of these -- as you said, they're all such extremely noticeable periodic phenomena. According to GNU units, the day is

  24 hr = 86400 s

(which is a historical basis for defining the second, though not the method we officially use now because we have clocks that are more consistent than the Earth's rotation), while the lunar month is

  29 days + 12 hours + 44 minutes + 2.8 seconds = 2551442.8 s

and the tropical year is

  365.242198781 day = 31556926 s

If we think of a month as a fixed whole number of days, and/or think of a year as a fixed whole number of months, and/or think of a year as fixed whole number of days, it's been known for thousands of years that we're eventually going to get in trouble astronomically and require some adjustments.

Indeed Plato considered astronomy to be the highest form of knowledge. To him it was superior to plane geometry (two-dimensional mathematics) and solid geometry (three-dimensional mathematics) since it was four-dimensional mathematics --- the study of the motion of three-dimensional objects through time.

I thought that the ancient Greeks considered the sky to be a 2 dimensional object: the inside of a sphere.

That conception was much older, probably in the early Archaic Age. By Plato's day it was believed that the planets were different distances from the Earth with the Moon being closest, Saturn being the farthest, and the sphere of the fixed stars being beyond that.

it sure does look like such a thing

then again, everything we see is majoritarily 2 dimensional (with a whisper or soft hint of depth)