What physically determines the point-set topology of a spacetime manifold?

There's no need to define the topology of the manifold from the metric. While a nice feature, the topology of the manifold is defined primarily by its atlas, which, from a physical perspective, correspond to the coordinates. A spacetime with a set of coordinates $\{ x^i \}$ will have a topology defined by the mapping of open sets from $\mathbb{R}^n$ to the manifold via the chart $\phi$.

If you wish, though, there are some things in general relativity that do define the spacetime topology.

A common basis of the spacetime topology is the Alexandrov topology. If your spacetime is strongly causal, the Alexandrov topology is equivalent to the manifold topology. Its basis is defined by the set of causal diamonds :

$$\{ C | \forall p, q \in M, C = I^+(p) \cap I^-(q) \}$$

It's easy to find counterexamples (the Alexandrov topology is just $\varnothing$ and $M$ for the Gödel spacetime), but if it is strongly causal, that will give you back the manifold topology.

There are lots of different possible ways of defining a manifold, some of which are not quite equivalent but all of which are equivalent for physics purposes. E.g., you can define a manifold in terms of a triangulation.

You could just start with the manifold, say defined using a triangulation. Then it has a definite topology, and only after that do you need to worry about putting a metric on it.

If you use the definition of a manifold in terms of a chart with smooth transition maps, then you get a topology for free from the charts. I think this is essentially what enumaris is saying.

But we should also be able to talk about these things in a coordinate-independent way. A metric can just exist on a manifold, regardless of whether the manifold was ever defined in terms of any coordinate charts. Then I think you still get a topology induced by the metric. This is because the metric defines geodesics, and it also defines affine parameters along those geodesics. So in your example of sending a photon to the Andromeda galaxy, the photon travels along a geodesic, we can define an affine parameter, and we can tell that the emission and reception of the photon do not lie in an arbitrarily small neighborhood of one another, because they lie at a finite affine distance.

I don't know about "physically" what defines open sets since open sets are a (afaik) purely mathematical construction, but what defines the open sets on the spacetime manifold is simply the open sets in $\mathbb{R}^4$. Open sets in $\mathbb{R}^4$ gets mapped to open sets in the manifold by definition. The topology of manifolds is induced naturally this way.