When to use which representation for an electric field In class we covered three types of possibilities to evaluate the electric field for static problems. Unfortunately, most physics textbooks cover these ways without addressing the question of applicability when they introduce these equations due to historical motivation.
One possibility is Gauß's law:
$$\int_{\partial V} \langle E, n \rangle dS = \int_V \frac{\rho}{\varepsilon_0} dx$$
This can only be used, when $E$ does by geometrical consideration not depend on the variables that the surface integral depends on(like in spherical symmetry, when it is not a function of the angles, but just of the radius), then it can be used to determine the electric field by a charged sphere. What we can get then, is an electric field on the surface of some volume that contains a particular charge.
Then we have:
$$ E(x) = \frac{1}{4 \pi \varepsilon_0} \int_{\mathbb{R}^3} \frac{\rho(x')(x-x')}{||x-x'||^3} dx'.$$
This is more general in the sense that it does not require any symmetry or reduction to charges inside some volume. 
And finally we have $$\Delta \phi(x) = -\frac{\rho(x)}{\varepsilon_0},$$
which is the most general way to think about electrostatic problems and contains (assuming that the solution fulfills all boundary conditions) all information.
I hope my understanding was correct so far. In that case, my question is:
How do I distinguish boundary value problems from problems where the first two equation apply?
Many problems are like: Given a sphere with a particular charge distribution, calculate the field everywhere around! What does this mean, does this mean that I should think of this as a ball of charges assembling in vacuum and the material around being vacuum too? Cause if I assume that this is a metallic sphere or a sphere made out of some material which is different from the material (probably varcuum) around, then this would give me an interface which means that my first two equations don't apply anymore, as I would get boundary conditions. ( I am talking now about this in a strict theoretical sense, maybe they would give me the right result, but for the wrong reasons). Or do these two equations also apply, if the ball is an insulator?
 A: 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 Kellogg, 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, Gauss'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 Gauss's differential equation, one of Maxwell's equations, $$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}.$$ This is also a general equation in a sense but Helmholtz'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 Poisson equation $\nabla^2 V = - \rho/\epsilon_0$ with appropriate boundary conditions.
This seems to be equivalent to the previous derivation due to the Helmholtz theorem (the condition of Hemholtz's theorem $\lim_{|\vec{x}| \to \infty} V(\vec{x}) =0$ is Dirichlet B.C. for the Poisson equation.).
Another way to derive Coulomb's law is by starting with the Maxwell equations and employing the Helmholtz 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 Coulomb's law becomes $$\vec{E}(\vec{x}) =\frac{1}{4\pi \epsilon} \frac{q_f (\vec{x} - \vec{x}')}{| \vec{x} - \vec{x}' |^3},$$ Poisson's equation becomes $\nabla^2 V = - \rho_f/\epsilon$, etc. However, if $\vec{\nabla} \times \vec{P} \neq \vec{0}$, which is the case on the boundary between different dielectrics (discontinuous $\epsilon$), then  Maxwell's equations in dielectric should be solved and there is no Coulomb's law for $\vec{D}$ that covers the entire space, which is another reason that Maxwell's equations are considered fundamental instead. Another way to see this is that a charge inside dielectric induces polarization charge, which makes the process non-linear, depending on the boundary between the dielectrics, therefore the principle of superposition cannot be used to find $\vec{D}$ in the entire space. Thus, the most general equations in presence of dielectric are Gauss's law and Poisson's equation.
General references: Griffiths, Electrodynamics, Zangwill, Modern electrodynamics, Jackson, Classical electrodynamics. 
A: 
I hope my understanding was correct so far. In that case, my question is: How do I distinguish boundary value problems from problems where the first two equation apply?

In static case, all those formulae are valid at the same time. You may use whichever you want. Based on the formulation of the problem, I would choose whichever is best to solve it.
You will find which method is best for what gradually by doing exercises and harder problems in electrostatics.
