Is it possible the space-time manifold itself could stop at a black hole's event horizon?

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.

The fundamental reason that we do relativity on a manifold, not a manifold-with-boundary, is the equivalence principle. One way of stating the e.p. is that every region of spacetime is locally describable by special relativity. That is baked into the structure of GR and the Einstein field equations, and it would be violated at a boundary.

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.

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 in classical GR. Nothing special happens, locally, at the event horizon. The event horizon is a set of points defined only in relation to distant points.

Historically, the misbehavior of the Schwarzschild metric, expressed in Schwarzschild coordinates, was not clearly understood at first. Later people realized that it was only a coordinate singularity. In GR, we aren't normally interested in spacetimes that are not maximally extended. When a spacetime has a proper extension, that is usually interpreted as meaning that something has just been artificially deleted from it. For example, you can take Minkowski space and delete a point, or delete everything at $t\ge 0$, but this is considered the kind of silly, artificial example that we want to rule out. We only want to talk about incomplete geodesics if the geodesics end at a singularity (a real singularity, not a coordinate singularity).

The reason that proposals such as firewalls are so radical is that they violate the equivalence principle. When, for example, people attempt to do semiclassical gravity and wind up with a prediction that something diverges at the event horizon of a black hole, it's a sign that their technique for doing semiclassical gravity isn't working properly. They can try to do things like renormalizations in order to get rid of this unphysical behavior. The basic problem is that semiclassical gravity lacks any clearly defined foundational principles. We have no reason to think that the techniques people use are valid approximation schemes.

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 Is a black hole singularity a single point?

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?

Such models are being considered. In the recent 20 years there has been a great interest in such models in connection with Hořava–Witten theory:

• Hořava, P., & Witten, E. (1996). Eleven-dimensional supergravity on a manifold with boundary. Nuclear Physics B, 475(1-2), 94-114, doi, arXiv.

There the strongly coupled limit of the $E_8 \times E_8$ heterotic superstring is identified with M−theory compactified on a $S_1/\mathbb{Z}_2$ orbifold (a line segment with two endpoints) with $E_8$ gauge fields on each orbifold fixed plane. Those fixed planes are two parallel boundaries of the 11 dimensional space-time: Hořava–Witten domain walls. Of course, the original scenario provides a new compactification method, rather than a way to include boundaries in the usual 4D spacetime. So in a sense, this model is somewhat similar to a braneworld cosmologies.

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

One of the insights from Hořava–Witten models (and quite a lot of earlier works) is that there must be dynamical degrees of freedom living on such boundaries. So the geodesic incompleteness that would generally be a feature of manifold with boundary does not lead to indeterminism: if something 'flies' into the boundary there must be equations telling us which boundary degrees of freedom would be excited, how the boundary is deformed etc. This is usually done by specifying both bulk and boundary actions for the theory. For example, we could construct the pair using supersymmetry:

• Belyaev, D. V. (2006). Boundary conditions in supergravity on a manifold with boundary. Journal of High Energy Physics, 2006(01), 047, doi, arXiv.

The takeaway is that we cannot just "punch holes" in spacetime manifold, we must do it consistently so that both bulk and boundary are solutions for corresponding equations of motion.

could the event horizon of a black hole be just such a boundary

In black hole physics there is membrane paradigm, a model that considers a black hole as a thin membrane microscopically close to the black hole's event horizon.

• Thorne, K. S., Price, R. H., & MacDonald, D. A. (Eds.). (1986). Black holes: the membrane paradigm. Yale university press.

Such membrane would have physical properties, mass, temperature, electrical resistivity etc. From the point of view of this model, there is nothing 'inside' of this membrane, everything that falls into the black hole is incorporated into the membrane. Of course, this is usually considered a simplified visual and computational aid.

However, there are models of black holes (or to be more precise 'exotic compact objects', ECO, since many of those models do not have a horizon) that assume the existence of new physics microscopically close to the (would be) horizon. Since the ringdown modes from mergers of black holes/ECOs would contain information about this physics, there is hope that LIGO (or next generation of gravitational wave detectors) could offer some observational evidence of this new physics.

One interesting analysis:

• Abedi, J., Dykaar, H., & Afshordi, N. (2017). Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons. Physical Review D, 96(8), 082004, doi, arXiv.

finds 'tentative evidence for Planck-scale structure near black hole horizons'.

As a sample of models for the ECO that are being considered have a look at

• Maggio, E., Pani, P., & Ferrari, V. (2017). Exotic compact objects and how to quench their ergoregion instability. Physical Review D, 96(10), 104047, doi, arXiv.

where authors consider ECO model which is Kerr metric for $r>r_0$ and a reflective membrane at some $r=r_0$. This model most closely resembles 'horizon as a boundary' suggestion.