General relativity says space-time is a $4$-dimensional manifold which may have non-zero global curvature. Now if we take a random curve or surface or $n$-fold, it may fail to be a manifold because it self-intersects (and the neigborhoods of a self-intersection point are not homeomorphic to $\mathbb R^n$). Are there any reasons this does not happen with space-time? What would be the physical meaning of the self-intersecting locus of the universe?
Are there any reasons this does not happen with space-time?
You seem to be assuming that curves and surfaces are always embedded within a higher-dimensional space. That's not the case, and we have no evidence for any higher-dimensional space in which our universe is embedded. If it's not embedded in any higher-dimensional space, then this whole issue doesn't even arise.
It's also problematic to try to make physical theories that don't live in a nice well-behaved manifold. For instance, you could try to generalize to a manifold with boundary, but then the problem is that experiments done on the interior can never reveal to you what the laws of physics should be at the boundary. Furthermore, we prefer theories that have uniqueness and existence of solutions for Cauchy problems, but usually you will not get those properties if you tamper with the topological properties and make something that's not a manifold.