1. Do you have a position in the philosophical debate about whether "general covariance" has a "physical" meaning, or is merely a property of the mathematical structure of the theory?

Yeah, I recall looking into this early in my grad studies. I eventually realized that the only content of it is diffeomorphism invariance, i.e. that one should be able to uniquely map tensor fields to spacetime points. The coordinate representation of these fields depends on the choice of coordinates, but the fields themselves do not. In that sense the principle simply states that the relation spacetime manifold -> tensor field is a function (surjective map). For example, there is a unique metric tensor at each spacetime point (which, incidentally, precludes traveling into one's past).

I would also like to mention that the debate "about whether "general covariance" has a "physical" meaning, or is merely a property of the mathematical structure of the theory" makes no sense to me as an instrumentalist (I consider the map-territory moniker an oft convenient model, not some deep ontological thing).

[I]f the whole universe was rotating around you while you stood still, you would feel a centrifugal force from the incoming gravitational waves, corresponding exactly to the centripetal force of spinning your arms while the universe stood still around you.

This is false, as far as I can tell. The frame dragging effect is not at all related to gravitational radiation. The Godel universe is an example of an extreme frame dragging due to being filled with spinning pressureless perfect fluid, and there are no gravitational waves in it.

it implies that the electromagnetic force (which is what causes your voluntary movements, such as "spinning your arms around") can be transformed into gravity by a change of coordinates?

Well, yeah, this is an absurd conclusion. The only thing GR says that matter creates spacetime curvature. A spinning spacetime has to correspond to spinning matter. And spinning is not relative, but quite absolute, it cannot be removed by a choice of coordinates (for example, the vorticity tensor does not vanish no matter what coordinates you pick). So Mach is out of luck here.