Can 2 different events in spacetime be connected by both a time-like and a space-like curve?
With flat Minkowski metric I don't think so, because any global orthonormal map $(x,y,z,t)$ of spacetime has its coordinate $t$ strictly increasing for all timelike (continuous) curves.
I'm not so sure with a general curved metric. Are there sufficient conditions on it so that these weird connections cannot happen?
Is it because the light cone of an event $E$ (the collection of all light-like geodesics going through $E$) separates spacetime in 3 connected components : past, future and space?