Algorithmic problem of regular, context-free, and recursively enumerable languages
Both questions are undecidable.
Notice that the empty language is both regular and context-free.
Thus we can state that the problem of determining if $L_1 \cap L_2$ is context-free is at least as hard as determining if $L_1$ is context-free (and similarly determining if $L_1 \cap L_3$ is a regular language is at least as hard as determining if $L_1$ is a regular language). By Rice's Theorem any non-trivial property about a Turing-Machine is undecidable and thus your two questions are undecidable.
For a detailed proof/answer to your second question look at this post.
IT shows how you would apply Rice's Theorem in your cases.