Crossing a manifold with Z2 (under the discrete topology) gives you a disjoint union of two copies of the original manifold. So, (1) it's certainly not connected, because Mx{0} and Mx{1} are open, disjoint, and cover the space. And (2) it's locally connected iff the original manifold is. And (3) it's a manifold in the obvious way, just as any disjoint union of (like-dimensional) manifolds is.