> If a photon follows a curve around a large mass, it does so because that curve is the shortest path through spacetime, if I am not mistaken.
Correct!
> Maybe I am missing something, but how could a quantum theory of gravity explain curvature of spacetime? […] how would a theoretical graviton explain this effect?
Note that a quantum theory of gravity need not involve gravitons. The latter only appear in approaches that assume that gravity is a more or less ordinary [string] quantum field theory (QFT). Personally, I've never found this very convincing. Neither did Hawking by the way, who once wrote
> It would be rather boring if this were the case. Gravity would be just like any other field. But I believe it is distinctively different, because it shapes the arena in which it acts, unlike other fields which act in a fixed spacetime background.[0]
Anyway, assuming we're indeed looking at a quantum theory of gravity that involves gravitons, let's tackle your other questions:
> Would it interact with the photon to carry over a force?
Yes, the graviton would interact (very weakly) with all particles we know, including itself.
> And if so, how would it "know" the direction of said force (wouldn't it be itself subject to gravity?).
My first, rather theoretical answer would be: This is not a well-defined question. After all, gravity as a force is a Newtonian concept which does not exist in General Relativity. As you already said, in Relativity matter is "guided" by the curvature of spacetime, not by a force. Moreover, in quantum mechanics Newtonian forces don't (really) exist, either, and while people often speak of "forces" as if they were Newtonian, these forces are really mediated by bosons.
My second, more direct answer would be: In two-particle interactions in quantum field theory, the outgoing particles usually leave the scenery at a different angle (and momentum) than their initial one. So the photon might get deflected by interacting with the graviton.
> Or would [the graviton] interact with space itself?
As already mentioned, the graviton would interact with itself in the same way as gravitational waves can interact with one another. As for space(-time) itself, however, note that the aforementioned "theories involving gravitons" usually assume that spacetime itself is flat (i.e., it is your run-of-the-mill Minkowski spacetime) and that gravitons are just another quantum field on that (so-called) background spacetime. As I personally don't like this idea a lot (see above), I can only guess as to whether its proponents really think spacetime is flat or whether they only assume this to simplify things. After all, we have no idea how to do QFT on curved spacetime in a mathematically precise way. (In fact, we don't know how to do QFT on a flat Minkowski background in a mathematically precise way, either.)
[0]: Hawking: The Nature of Space and Time, https://arxiv.org/abs/hep-th/9409195v1