Which closure properties are always valid between regular, context-free and non context-free languages?
I am afraid that there is no better general result than the obvious one.
Assume that $\mathcal K$ and $\mathcal L$ are two families of languages with $\mathcal K\subset \mathcal L$ such that $\mathcal L$ is closed under the operation $\circ$. Then for $K\in\mathcal K$ and $L\in \mathcal L$ we have $K\circ L\in \mathcal L$.
This is obvious, as both $K,L\in \mathcal L$. Usually we can choose simple examples that we can't do better. For instance, choose a &quot;difficult&quot; language in $\mathcal L$, so $L\in \mathcal L \setminus \mathcal K$. Then $L=\varnothing \cup L$, $L=\Sigma^*\cap L$, and $L=\{\lambda\}\cdot L$ show that we cannot do better than $\mathcal L$.