> is it permissible for a space-time manifold in general relativity to have an edge in the same sense that a piece of paper (a 2D manifold) has an edge?

Not really. The Einstein field equations only make sense at a point that has an open neighborhood of spacetime surrounding it, so we can only apply them on a manifold, not on a manifold-with-boundary. We do sometimes talk about a manifold-with-boundary in GR, but usually the context is that we're describing idealized points and surfaces that have been added to the spacetime, such as $\mathscr{I}^+$ or $i^0$. These are like vanishing points in perspective art. They are not actually part of the spacetime.

> if so, is it permissible for it to contain a "hole" in the same sense as if you punched a hole in said sheet using a hole punch

GR doesn't impose any constraints on the topology of spacetime, so you can have holes. However, a hole does not imply a boundary in topology. If you take the Cartesian plane and remove the closed unit circle $r\le1$, you get a manifold, not a manifold-with-boundary.

Note that if you look at the Penrose diagram for a black hole, and you remove the region inside the event horizon, it doesn't change the topology. The topology is still the trivial topology.

> if that is so, could the event horizon of a black hole be just such a boundary (or connected set of boundary points) of the spacetime manifold?

There is no physical motivation for doing this. Nothing special happens, locally, at the event horizon. The event horizon is a set of points defined only in relation to distant points.

> I note that it appears the regular singularity of an ordinary black hole is (at least from what I gather in readings) a boundary of dimension 1

Not true. There is no standard way to define its dimensionality. See https://physics.stackexchange.com/questions/144447/is-a-black-hole-singularity-a-single-point