How exactly does air "absorb" light? What happens to it? Where does all that energy go?
It gets absorbed as heat, which is later dissipated as infrared back into space.
How does the air know from which direction the light is coming from?
As I said, the air is magnetoresonant horizontally. It will therefore only absorb the vertical component of the light's motion.
Why isn't this true of all light? Why isn't this true of street lamps?
Why would you assume that it isn't?
Why hasn't this phenomenon been observed in experiments designed to test the visibility of air (and various other gasses)?
You'd have to ask the experimenters that.
Most interestingly, if light from above is more readily "absorbed" why isn't the sky directly above my head a darker shade of blue?
Because it's closer to you. Ordinarily, it would be brighter, just as the Sun would overhead. This absorption causes it to appear more or less the same shade.
And last of all, what makes this theory "obvious" or "simple"?
The fact that if the Sun would be brighter when overhead, and it is observed not to be, then
something must be getting in the way of the light.
Then you're contradicting the magnification effect of Air to account for the sun appearing to retain it's shape and size during the day even though it's moving further away from us at a constant altitude.
Air doesn't magnify light in this way, and even if it did, that would have nothing to do with its effect on the intensity of the light.