Question

If coulomb's law involved $1/r^3$ instead of $\left(1/r^2\right)$, would Gauss's law still be true?

Answer 1

In this case the the electric field due to point charge $q$ is $E=\frac{kq}{r^2}$

Now consider a gaussian surface as a sphere surface of radius $r$

The flux through the surface is $\varphi =\oint E.dS=\left(kq/r^3\right)\oint dS=kq/r^3\times 4\pi r^2=\frac{4\pi kq}{r}$

Thus this flux is dependent on $r$ but from Gauss's law the flux depends only on the charge inside the surface.