Can the expected frequency of all sizes of impact craters on Earth be calculated based on the preservation of impact craters seen on the Moon? Taking into account the difference in gravity and atmosphere of the Earth and Moon, and assuming a similar age for both, it should be possible to come up with a reasonable estimate of how many meteorites, etc have hit the earth and their size distribution.
The simple answer to your question is, simply, yes.
All objects in the inner solar system is generally assumed to have been impacted by the same population of impactors which is mostly comprised of asteroids, and perhaps up to 10% comets. The outer solar system likely has a much larger percentage of cometary impactors.
Of the five main inner solar system terrestrial bodies, the Moon and Mercury have the most preserved crater record, idealized by a death of volcanism early in their histories, no atmosphere to speak of, and therefore craters are best preserved there. Mars is second-best (some atmosphere, localized volcanism over the last few billion years, and water in the first ~billion years). Venus is a mess for cratering (entire planet resurfaced ~400-800 million years ago plus a thick atmosphere that prevents craters <3-5 km from forming), while Earth has the most modified crater record.
With the moon next door, it is our best historical record for what also likely hit Earth.
The factors that will alter the lunar crater population from what would have formed on Earth are primarily gravity, surface type, and atmosphere. Atmosphere will not only screen out the smallest impactors (and hence not make craters), but it will also fragment less competent objects, changing what could have made a single large impact on the moon into something that will more likely make numerous smaller, clustered craters on Earth (from the surviving fragments). A larger surface gravity will tend to decrease the size of the final crater caused by a given impactor, but the dependence is small (difference of around 35% between Earth and the Moon despite the factor of 6 difference in gravity).
Surface type is much less well understood in the cratering community. Earth has stronger, denser crust than the Moon, but it's also ~70% covered by water -- the surface of Venus is like being under 1 km of water on Earth, so any ocean >1 km deep is going to prevent craters $\lesssim 3-5$ km from forming.
These all combine for the much longer and more complicated answer of, "yes, but it's hard to figure out." There are lots of knobs in the models that we still don't know how to correctly turn, but we can, to first-order, use the lunar crater population and rate to estimate the population of craters that should have formed on Earth. At the time of writing this, there are 184 confirmed terrestrial impact craters, which is certainly a tiny fraction of a percent the number that have formed over Earth's history.
Yes. By looking at a nearby object that hasn't been subject to erosion and has been subject to the same position around the sun, you could estimate the distribution of impacts on the Earth in time and in size. The moon is excellent for this purpose.
Here's an article touching on the subject: http://adsabs.harvard.edu/full/1975LPSC....6.2597N