I think the answer to the second part (both $C_i$ strict) is negative.
Fix a finite group $G$ with a non-trivial 2-cocycle $\gamma \in H^2(G, \mathbb{C}^{\times})$. Let $C$ be the strict $\mathbb{C}$-linear monoidal category with simple objects $\{g: g \in G\}$ and tensor product $g \otimes h = gh$. Then the 2-cocycle $\gamma$ can be used to define a monoidal functor $(1_C, \gamma): C \to C$. This functor is not naturally isomorphic to a strict functor (the only possibility is it is isomorphic to the (strict) identity functor, but this cannot be the case since $\gamma$ is non-trivial).