Is it true that Macrorealism can be shown false? http://en.wikipedia.org/wiki/Leggett-Garg_inequality
As you can see by the link above, it claims that if the violation of the Leggett–Garg inequality can be demonstrated on the macroscopic scale, it would challenge the notion of realism that the moon is there when nobody looks.
How is that even possible given that when you get larger than a molecule, the quantum wavelength shrinks to insignificance?
According to the link, "they" are even trying to actively falsify macrorealism by employing superconducting quantum interference devices.
 A: In theory yes, but in practice no. Showing it "false" would not be feasible.
"The moon won't exist if we stop looking at it".
It will still exist, just as an atom still exists when not "measured", however it's exact position and state will not be as precisely defined. How "not precise"? The Schrodinger equation solutions show a wave packet spreading out as time goes on. The failure of "realism" means, in this case, that the true position is not just a hidden variable that we choose to ignore. However, the heavier the particle/wave, the less it spreads out. This spreading effect is negligible for a dust grain let alone the moon, so it's location will still be very "real" and precise unless it stays "unobserved" far longer than proton decay timescales.
Could we entangle two atoms and violate local realism on a lunar distance scale? Sure! We just need a larger apparatus to isolate our particles, and more precise aiming so it doesn't hit the walls. The results would mirror smaller-scale experiments.
Could we entangle an atom in lunar soil moon with one of ours on earth? No. The atom is in contact with the rest of the planet/moon. The quantum state of the entire planet needs to be solved in order to know that your (otherwise random) data would demonstrate entanglement. Also, "not observing" it means not a single photon absorbed/emitted from the entire planet (let alone gravitational interaction). 
