I think you will find that asteroids do not have moons.

Yes, moons of Mars

Yes. Afew asteroids have smaller asteroids orbiting them as moons. The best known case is the asteroid Ida and its moon Dactyl.

No. An asteroid can get captured by an asteroid and become a moon, but not the moon. Our moon is too large to be considered an asteroid.

Both asteroids and moons can be rocky. An asteroid orbits the Sun while a moon orbits a planet.

Mars has two moons (Deimos and Phobos) that are most likely former asteroids.