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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. In fact, if you remove the interior of the black hole from the Penrose diagram for an eternal black hole, you just get the Penrose diagram for Minkowski space. If your intuition is that removing the interior leaves a topological hole, it's probably because you're incorrectly imagining the event horizon as a timelike surface. It's a null surface (and the singularity is spacelike).

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.

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 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.

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.

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 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?

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. In fact, if you remove the interior of the black hole from the Penrose diagram for an eternal black hole, you just get the Penrose diagram for Minkowski space. If your intuition is that removing the interior leaves a topological hole, it's probably because you're incorrectly imagining the event horizon as a timelike surface. It's a null surface (and the singularity is spacelike).

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.

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).

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?