Deciding whether the language of a CFG equals some specific regular language
It depends on your choice of $A$. For example, the problem $\{\langle G \rangle \mid L(G) = \Sigma^* \}$ is undecidable, while $\{\langle G \rangle \mid L(G) = \phi \}$ is decidable.

Also, the two languages that you have given (the one with $A$ and the other with $\overline{A}$) are equivalent since regular languages are closed under complementation.