Why charges must have a spherically symmetric distribution?
You can see in the Wikipedia article there is a section "Deriving Coulomb's Law from Gauss's Law". Gauss's law is a fairly fundamental result in multivariable calculus, not limited at all to the field of physics, so this is the direction we should be going when deriving one law from the other.
Notice that this derivation depends on assuming the source is spherically symmetrical. Without this assumption (or pre-condition on the physical situation to be analyzed), Gauss's law doesn't give the result that we call Coulomb's law and Coulomb's law is not valid.
That said, we can in fact use Coulomb's law to analyze non-symmetric systems. We just have to use the section of the Wiki article that says,
A system ''N'' of charges $q_i$ stationed at $\mathbf{r}_i$ produces an electric field whose magnitude and direction is, by superposition
$$\mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^N q_i
\frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3}$$
For systems with continuously distributed charge rather than a finite number of point charges, we can extend this by taking infinitesimal elements of charge and turning the summation into an integral.
$$\mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \int_{R} q'(\mathbf{r'})
\frac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3}d\mathbf{r'}$$
Where $R$ is the region containing charge and $q'(\mathbf{r})$ is the charge density at point $\mathbf{r}$.
