Do celestial bodies have many asteroids orbiting them?
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?