$R^2f_*\mathbb{Z}$ is isomorphic to the constant sheaf $H^2(X_0,\mathbb{Z})$

Question

Let $f:X \rightarrow S$ be a smooth proper family of K3 surfaces and let $X_t=f^{-1}(t)$. Let's suppose $S$ is a disk in $\mathbb{C}^n$. I know that the Betti/Hodge numbers are constant, so the direct image $R^2f_*\mathbb{Z}$ is locally free and I can study it as a vector bundle.

1. What are the fibers of this vector bundle? Intuition says the fibre at $t \in S$ is $H^2(X_t,\mathbb{Z})$, but how do I prove it?
2. Why is $R^2f_*\mathbb{Z}$ isomorphic to the constant system $H^2(X_0,\mathbb{Z})$?
3. What about the sub-bundle $f_* \Omega_{X/S}^2 \subset R^2f_*\mathbb{C} \otimes_\mathbb{C} \mathcal{O}_S$? What are its sections?