Are there any applications of set theory in physics, which are of interest to physicists, and have important applications in physics?

I'm a physicist. Any answer to this question is going to depend completely on your definitions of "application" and "set theory." If you consider only some trivial corner of naive set theory, and count cases where its use is purely a matter of convenience or ease of notation, then certainly there are many applications. For example, we often solve quadratic equations in physics, and it's convenient to talk about the set of real solutions.

But every physics experiment that has ever been done was performed with finite physical and computational resources, which means that all of our experience of physics can be described within finitism. There are even rigorous arguments (Krauss 1999) that, given the cosmological facts we observe, any hypothetical future physical process will be able to harness only finite energy and finite computation. (This is nontrivial; before the discovery of dark energy, the opposite conclusion was reached by Dyson.)

Because of these physical limitations, it is not possible, even in principle, for us to measure an irrational number or to demonstrate affirmatively that spacetime has the structure of a manifold.

I only have access to the abstract of the Augenstein paper, but it seems to argue that "physical reality," specifically quantum mechanics, provides direct realizations of set-theoretical constructs, including "individual axioms" of ZFC. This sounds bogus to me on both physical and mathematical grounds.

Physically, Krauss's result tells us that we can never harness more than a finite amount of energy, and via the de Broglie relation, this puts a a cutoff on the wavelengths that we will be able to probe. The region within our cosmological event horizon will always be finite as well. The combination of these two cutoffs means that any quantum-mechanical experiment, even in principle, can be described by a finite-dimensional Hilbert space.

Mathematically, the vast majority of mathematics is carried out without any consideration of any underlying foundational issues, such as the use of ZFC as opposed to some other framework. The sphere within which physicists operate is even more restricted than that of normal mathematics.

We can also consider the role of computation, through which a physical machine (such as a computer, a brain, or a slide rule) can prove things about mathematics. I can use an analog computer to compute the square root of 2, e.g., by constructing two pendulums with lengths in a 2:1 ratio and measuring the ratios of their periods. If my analog computer says that the second decimal place of the decimal expansion of $\sqrt{2}$ is a 1, and your computation says that it's a 3, then my physics experiment has successfully demonstrated something to you about your mathematical theory. If the axioms of ZFC were to be directly realized in physical experiments, as Augenstein seems to propose, then I ought to be able to do the same kind of thing with ZFC. Does anyone really expect that a physicist will do an experiment that will prove the axiom of choice?