The most generally applicable equation of electrostatics is the Coulomb field of a point charge $q$ at position $\vec{x}'$ in vacuum, $$\vec{E}(\vec{x}) = \frac{1}{4\pi\epsilon_0} \frac{q(\vec{x} - \vec{x}')}{ | \vec{x} - \vec{x}'|^3}.$$ This is an experimentally derived equation, if we regard small charges as point charges. Your second equation can be derived from this by employing the principle of superposition, hence we have $$\vec{E}(\vec{x}) = \frac{1}{4\pi\epsilon_0} \int_D \frac{\rho(\vec{x}')( \vec{x} - \vec{x}')}{ | \vec{x} - \vec{x}'|^3} d^3x',$$ with $D$ a region that includes the charge distribution under consideration. The integral converges if $D$ is finite, $\rho$ is bounded and the integrand becomes infinite at most once, according to <PERSON>, Foundations of potential theory. In the case of infinite $D$ in general the integral does not converge. There are some distributions that allow it to converge, such as the homogeneously charged, infinite plane or wire; they are highly symmetric.
From the Coulomb integral, if it converges, <PERSON>'s integral law and $\vec{\nabla} \cdot \frac{\vec{x} - \vec{x}'}{| \vec{x} - \vec{x}'|^3}=4\pi \delta(\vec{x} - \vec{x}')$, we obtain <PERSON>'s differential equation, one of <PERSON>'s equations, $$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}.$$ This is also a general equation in a sense but <PERSON>'s theorem states that certain conditions must be met (localized sources) in order for $\vec{E}$ to exist and be defined uniquely by this equation (here we assume $\vec{\nabla} \times \vec{E} = 0$, we're doing electrostatics). The same conditions imply the existence of the potential of the distribution, but $\vec{\nabla} \times \vec{E} = 0$ is sufficient to deduce $\vec{E} = - \vec{\nabla} V$ without them (it is noteworthy that for a homogeneously charged, infinite plane $V$ diverges but the previous equation yields the correct field regardless!).
Another way to obtain the potential of the distribution is to solve the <PERSON> equation $\nabla^2 V = - \rho/\epsilon_0$ with appropriate boundary conditions. 418
This seems to be equivalent to the previous derivation due to the <PERSON> theorem (the condition of <PERSON>'s theorem $\lim_{|\vec{x}| \to \infty} V(\vec{x}) =0$ is Dirichlet B.C. for the <PERSON> equation.).
Another way to derive Coulomb's law is by starting with the <PERSON> equations and employing the <PERSON> theorem for localized sources, i.e. ones for which the field vanishes faster than $1/r$ as $r$ goes to $\infty$. This derivation has the advantage that both integrals always converge and the disadvantage that it excludes distributions of charge that extend to $\infty$, which are not physical but are used to extract essential properties of more complicated geometries. A work-around for this limitation is to consider a sequence of localized distributions that tends to infinite extent.
Finally, these equations still apply in presence of dielectric other than the vacuum, but proper assumptions must be made beforehand. For example, if the dielectric is linear, isotropic and homogeneous, then $\vec{D} = \epsilon \vec{E}$ and <PERSON>'s law becomes $$\vec{E}(\vec{x}) =\frac{1}{4\pi \epsilon} \frac{q_f (\vec{x} - \vec{x}')}{| \vec{x} - \vec{x}' |^3},$$ <PERSON>'s equation becomes $\nabla^2 V = - \rho_f/\epsilon$, etc. 469