Coulomb's law cannot be applied to moving charges.
The Lienard-Wiechert potentials have to be used.
Since div E=4pi rho in all cases, Gauss's law still applies.
However since the symmetry is lost, Gauss's law just gives the integral of E over a closed surface and can't be used to find E(r).
Well,I found it in a text I mentioned.I think we may think this way:the charge inside the closed surface will be static or moving.Whatever the case,the fluxes are electric as well as magnetic.Gauss's law still holds because, magnetic flux out of a closed surface is zero.We do not find the B field when charges inside are in motion.
This is clear if we look at the Maxwell Equations. The curl of E is no longer zero, for the particle is moving and that makes the B changing with time. Thus the Coloumb's Law fails but div E is unchanging and Gauss's Law holds.