Yes, this is correct. It is generally true that the evaluation functors $\text{ev}_Z \colon [\mathcal{J},\mathcal{D}] \to \mathcal{D}$ for $Z \in \mathcal{J}$ jointly create limits (and colimits). So any limit in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ can be computed pointwise if and only if the pointwise limits exist in $\mathcal{D}$. In particular when $\mathcal{D}$ is complete, the limit defining $(F \multimap G)X$ in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ exists and is preserved by evaluation at $Z$.

Yes, this is correct. It is generally true that the evaluation functors $\text{ev}_Z \colon [\mathcal{J},\mathcal{D}] \to \mathcal{D}$ for $Z \in \mathcal{J}$ jointly create limits (and colimits). So a limit in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ can be computed pointwise if and only if the corresponding pointwise limits exist in $\mathcal{D}$. In particular when $\mathcal{D}$ is complete, the limit defining $(F \multimap G)X$ in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ exists and is preserved by evaluation at $Z$.