Why is charge $q$ symmetrically distributed? Simple question: Why is charge $q$ outside symmetrically distributed?
The material is a conductor.

 A: The charge is spherically symmetric because that ensures that the field will be zero in all the interior of the sphere (including inside the cavity) regardless of the interior shape of the conductor. If there was any asymmetry on the charge distribution you could prove that there exists an interior electric field. (provided no other charges exist than those on the conductor)
A: When we put charge to the point marked $q$ inside the cavity, both the inner surface of the cavity and the outer surface of the sphere will become charged by induction.
The distribution of charge on the outer surface is symmetrical as a result of what happens inside. The charge inside the cavity gets distributed in such a way, that it doesn't allow the field from $q$ to penetrate into the mass of the conductor (otherwise the charge inside the conductor would move -> to reach the equilibrium).
Therefore the distribution of charge on the outer surface is not influenced by geometry of the cavity or by the position of the charge $q$ inside the cavity.
