Inverse Homomorphisms and Kleene star
Since you solved the first question, let me answer the second one. If you don't mind, I will use $h$ instead of $h'$ for simplicity. Let $L$ be a regular language and let $K = h^{-1}(L^*)$. Since regular languages are closed under inverses of homorphisms, $K$ is regular. Moreover, if $u, v \in h^{-1}(L^*)$, then $h(u), h(v) \in L^*$ and hence $h(uv) = h(u)h(v) \in L^*$. It follows that $uv \in h^{-1}(L^*)$ and thus $K = K^*$. Consequently, $h^{-1}(L^*) = K^*$, which answers your question.