Answer by Donu Arapura for Hodge numbers of reduction mod $p$

For some explicit counterexamples to (2) and therefore also (4) again, see J. Suh,Plurigenera of general type surfaces in mixed characteristic, Compositio (2008).The only thing that you can say for sure is that $h^{pq}(\bar X)\ge h^{pq}(X)$ byupper semicontinuity of cohomology.