Results from a few years ago of Kennedy and Schafhauser characterize simplicity of reduced crossed products $A \rtimes_r G$, where $A$ is a unital C*-algebra and $G$ is a discrete group, under an assumption which they call vanishing obstruction. However, this is a strong condition that often fails, even in cases of $A$ being finite-dimensional and $G$ being finite. In joint work with Shirly Geffen, we find the correct two-way characterization of when the crossed product is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups which have polynomial growth. With some additional effort, we can characterize the intersection property for $A \rtimes_r G$ in the non-minimal setting, for the slightly less general class of FC-groups. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to generalize a result by Hamana for finite groups, and characterize when the crossed product $A \rtimes_r G$ is prime. All of our characterizations are initially given in terms of the dynamics of $G$ on $I(A)$, the injective envelope of $A$. This gives the most elegant characterization from a theory perspective, but $I(A)$ is in general a very mysterious object that is hard to explicitly describe. If $A$ is separable, our characterizations are shown to be equivalent to an intrinsic condition on the dynamics of $G$ on $A$ itself.