Does applying a homomorphism to the intersection of two CSLs yield RE languages?
This follows from closure properties of CSL: it's closed against intersection and $\epsilon$-free homomorphism, but not against (general) homomorphism.

Thus, $h(L_1 \cap L_2)$ is always context-sensitive if $h$ is $\epsilon$-free.

If $h$ is not, $h(L_1 \cap L_2)$ may not be. As an example, take $L_1$ and $h$ from the proof that CSL is not closed against general homomorphism, and $L_2 = \Sigma^*$.

Since CSL $\subseteq$ RE and RE is closed against intersection as well as (general) homomorphism, $h(L_1 \cap L_2)$ is indeed always in RE.