I'm wondering whether $\mathsf{PSh}(\{x\})$ or $\mathsf{Sh}(\{x\})$ are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the only degree of freedom except choice of singleton is the value on $\{x\}$. Still though, I'm not sure whether I'm not talking nonsense.
For presheaves, I really don't know at all..
Yes, if $\mathcal{C}$ is any category with a terminal object $1$, then $\mathsf{Sh}(\{x\},\mathcal{C}) \to \mathcal{C}$, $F \mapsto F(\{x\})$ is an isomorphism of categories.
On the other hand, there is an isomorphism of categories $\mathsf{PSh}(\{x\},\mathcal{C}) \to \mathsf{Mor}(\mathcal{C})$ which maps $F$ to the restriction morphism $F(\{x\}) \to F(\emptyset)$.
By the way, under these isomorphisms, the forgetful functor from sheaves to presheaves becomes $\mathcal{C} \to \mathsf{Mor}(\mathcal{C})$, $X \mapsto (X \to 1)$, and its left adjoint (sheafification) is simply $\mathsf{Mor}(\mathcal{C}) \to \mathcal{C}$, $(X \to Y) \mapsto X$.