It does! See, for example, http://en.wikipedia.org/wiki/Chicxulub_crater -- the asteroid that killed the dinosaurs left a very noticeable gravity anomaly, which helped us identify the site as an impact crater.

The size difference would be the main reason. On earth, the dense material is much deeper, so you'd need a lot more force to bring the it to the surface. The increased size and soft earth probably absorb more of the shockwaves as well. Having no atmosphere means that even smaller impacts will reach the surface as well.

There are probably some impacts that have caused this on earth, but the difference in gravity is probably so small, that it doesn't effect orbits. (We could probably measure it though).

It does, but a lot less: http://en.wikipedia.org/wiki/Geoid

Earth's radius is 3.7 times moon's; volume is 54 times. This difference may translate to the difference in phenomenon.