Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:

A) Is there a classification of the possible homeomorphism types of such $M$?

B) Is there a classification of the possible diffeomorphism types of such $M$?

Hitchin has proved that if $\vert \tau(M) \vert = \frac{2}{3}\chi (M)$ then $(M,g)$ is isometric to a K3 surface equipped with a Ricci flat metric. Here $\tau(M)$ denotes the signature of $M$ and $\chi(M)$ denotes its Euler characteristic. Therefore, through my two questions above I am wondering about the diffeo or homeo type of $M$ in the case $2\chi (M) + 3 \tau(M) >0$.

Alternatively, if $(M,g)$ is irreducible, simply connected and Ricci-flat four-manifold, Berger classification implies that its holonomy group must be either SU(2) or SO(4). If Hol($g$) = SU(2) then $(M,g)$ is a K3 surface equipped with a Calabi-Yau metric. Otherwise $(M,g)$ has generic holonomy. It is currently not known if there exist Ricci-flat compact manifolds of generic holonomy. In this context, questions A) and B) about concern the possible diffeo or homeo types of simply connected compact Ricci-flat four manifolds of generic holonomy. The previous remarks suggests a third question:

C) Is every Ricci flat metric on a K3 surface Calabi-Yau?

Thanks.