Relativity and quantum theory are currently irreconcilable
This is incorrect. Relativistic quantum field theory is almost a hundred years old, and there is nothing wrong with it. It is actually the "language" of particle physics! Difficulties arise when we try quantizing gravity (or general relativity, to be precise). Special relativity can be easily dealt with in quantum theory.
What keeps spacetime from just being the summation of all the quantum fields? The fields are scalar, and everywhere...could it be they're all just different degrees of freedom in spacetime?
In a sense, the "shape" of spacetime is given by the metric, which is a field. General relativity is a classical field theory, and in this sense spacetime is already a field. No need to "sum all quantum fields" or anything. The tricky bit about gravity is how to obtain a quantum description of the metric field. This may require introducing new fields (which appear in modified theories of gravity).
I know gravity doesn't play well with quantum theories because it doesn't seem to be able to be quantized. When, from a relativistic point of view, spacetime curves in the presence of gravity, aren't the electromagnetic fields within that portion of spacetime also curved from the point of view of an external observer?
The analog of "curved spacetime" for electromagnetism is merely a nonvanishing electromagnetic field. If you have a charge, the electromagnetic field near it is curved, but in a more abstract sense. Sometimes, the electromagnetic field is even called curvature (especially in more geometric approaches to gauge theory).
In other words, if you can't quantize gravity, can you topographically bend quantum fields? (or all of them all at once because they're just expressions of different aspects of the unified whole of spacetime?)
This is misleading. Quantum fields can be "curved" in the sense that they are nontrivial. For example, the electromagnetic field does not need to vanish. However, this has nothing to do with gravity. Gravity in general relativity is described by its own field, which is the metric tensor. Other fields do not "store" the gravitational degrees of freedom and cannot describe the behavior of spacetime on their own. The tricky thing about quantum gravity is exactly how to describe the quantum behavior of the metric field.
In particular, as mentioned in the comments, we know how to describe quantum fields in curved spacetime.
Remark: I'm thinking here about the standard model fields and general relativity. There are scenarios in modified gravity in which things can get more complicated, but those are more speculative (for instance, the Kaluza–Klein remark that appeared in the comments). Perhaps someone with more experience on those topics may later comment on these ideas.
