Is the question of whether the language of a DFA/CFG is equal to a particular set of string decidable?
Let's assume that $S$ is regular and given in some explicit form (DFA, regular grammar, regular expression, what have you). Given a DFA $D$, we can compute a DFA for $L(D) \triangle S$ (here $\triangle$ is symmetric difference) and check whether it is empty or not. This allows us to decide whether $L(D) = S$ or not.

Given a CFG $G$, we can compute a CFG $G'$ for $G \cap \overline{S}$, and then check whether $L(G') = \emptyset$. This allows us to check whether $L(G) \subseteq S$. However, in general we cannot check the opposite inclusion: determining whether $L(G) = \Sigma^*$ is undecidable (the universality problem). However, if $S$ is finite, we can check whether $G$ generates each of the strings in $S$, and thereby solve the problem.