Do all moons orbit their planets on the ecliptic? Do all moons orbit their planets on the ecliptic? Do they all follow this same process as the moon that orbits Earth, or is their some random distribution with some even having extreme polar orbits.
Of course not.

See Neptune's moons. Triton's orbit has a tilt of 157º from Neptune's rotation plane, which is in itself tilted 30º from Neptune's orbit, just 2º away from the Ecliptic.

Or look at http://en.wikipedia.org/wiki/Moons_of_Uranus