Do celestial bodies have many asteroids orbiting them? Are there statistics on the expected number of asteroids (of various sizes) to orbit planets and moons in our (or other) solar systems?

Do we know of other natural bodies orbiting the Earth (besides the Moon) or asteroids orbiting the Moon?

If not, why not? Though the likelihood of natural captures are low, it is my understanding that there are lots of asteroids (though, of course, all orbits will eventually decay).
The short answer is not very often.   Article on that here:  can moons have moons?

Nothing can orbit the Moon long term cause the Earth affects a moon/moon-satellite orbit and it draws tidal energy from it so an object would crash into the moon.

A 2nd moon around earth might be more stable than one around the moon, but I suspect, not long term stable.   If it fell into orbital resonance with our moon, like the inner 3 moons of Jupiter, I don't know, maybe, but I suspect not, cause the Moon's tidal force on a 2nd moon would (I suspect) be too great.

From time to time, the Earth captures an asteroid, but they usually don't stay for long:  http://www.scientificamerican.com/podcast/episode/earth-usually-has-second-tiny-tempo-11-12-27/

Mars' 2 moons and many of the gas giant planet's moons are captured asteroids or Kupier belt objects and if the earth didn't have our large moon, we might have a few captured asteroids in our orbit as well, so planets absolutely can capture asteroids, but the closer to the sun (Mercury/Venus), the smaller the planet's gravitational sphere's of influence and the less stable long term.

Curiously, some asteroids even have asteroids, and Pluto in particular is a rather strange system with a comparatively enormous and very close moon, Charon and 4 outer moons that orbit around Pluto and Charon.   you can't get objects orbiting the Earth-Moon system, cause that would extend past the stable part of the Earth's Hill Sphere.   Pluto's Hill Sphere is 'about 3 times the radius of Earth's because it's so far away from any large bodies.  

Generally, stable orbits only exist in the inner 1/2 to 1/3rd of a body's hill sphere, so those are the 2 factors to look at, tidal effects and hill sphere (or Sphere of influence, which is related to Hill Sphere).

Sphere or influence numbers for the 8 planets and Pluto:

https://en.wikipedia.org/wiki/Sphere_of_influence_%28astrodynamics%29

Related:  What is the difference between Sphere of Influence and Hill sphere?