Homomorphism Languages
Unfortunately both sides contain the empty string. As you say, the Kleene star of the empty set also contains the empty string: $\varnothing^* = \{ \lambda\}$. The Kleene star of a language $L$, equals $L^* = \cup_{i\ge 0} L^i$, where $L^0 = \{\lambda\}$ and $L^i = L \cdot L^{i-1}$ for $i\ge 1$.

On the other hand $h(\lambda) = \lambda$ for any morphism $h$, so $\lambda \in h^{-1}(K)$ whenever $\lambda \in K$. As $L^*$ contains the empty string this is the case here.

However, this does not mean the two sets are equal.
Consider $h:\{a,b\}^*\to \{a\}^*$ defined by $h(a) = a$ and $h(b) = aa$ and $L=\{a\}$.