For the question about homeomorphisms the answer is no, even if $X$ and $X'$ are algebraic surfaces.
In fact, in his paper [Orientation reversing homeomorphisms in surface geography, Math. Ann. 292 (1992)], D. Kotschick proves the following result:
Theorem. There exist infinitely many pairs of simply connected algebraic surfaces of general type which are orientation-reversing homeomorphic (with respect to their complex orientations), but not diffeomorphic.
He also makes a conjecture about orientation-reversing diffeomorphic algebraic surfaces. As I said in my comments before, by using Seiberg-Witten theory one proves that, given any diffeomorphism $\phi \colon X \to X'$ between two smooth $4$-manifolds, one has either $\phi(K_X)=K_{X'}$ or $\phi(K_X)=-K_{X'}$.
Kotschick's conjecture is therefore the following:
Conjecture. If two algebraic surface with finite fundamental group are orientation-reversing diffeomorphic, then they are homeomorphic to a geometrically ruled rational surface. In particular, they are simply connected.
I do not know the current state of this conjecture.