This is an equation designed to compare space with time, though. If we were comparing xyt against z (using whatever terminology) then I see no reason why it wouldn't work. (In the 'rotated' fly-space, for instance.)
I'm pretty sure it wouldn't work. dr^2 - dt^2 is an invariant because of the equations for coordinate transformations between different reference frames, which should be the same in rotated fly-space. If you construct a similar equation to compare z-distance with xyt-distance, you'll get a quantity that can change from reference frame to reference frame, regardless of how you view the space it's measured in.
You've asserted that something can hold still in space while moving through time, but can you actually show evidence of this ever happening? In order for something to be shown as not moving we'd first have to demonstrate a reference frame that itself is not moving. This is against relativity. We can say that two objects are moving in parallel at the same speed, but that only means that they're holding still in reference to one another. It doesn't mean that either of them is not moving in space.
I'm not trying to claim that anything is holding still in the absolute sense that you mean - as you seem to agree, movement isn't meaningful in any absolute sense. I am claiming that something can hold still in a particular reference frame - that is, that its spatial coordinates in that reference frame could be constant. That much seems clear to me; in a reference frame attached to my desk, my monitor is stationary.
If I restrict my attention to inertial reference frames - that is, reference frames that aren't rotating, accelerating, or breaking the speed of light - I can then make my claim that objects can be stationary in space but not in time. In short, it doesn't matter that I have no absolute coordinates, because my argument holds true in any set of relative coordinates I choose, as long as those coordinates correspond to an inertial reference frame.
It suggests to me that the fly interacts with the dimension of time the same way that we do. Just like a flatlander can't freely interact with the third dimension, we can't freely interact with the fourth. This does not effect the nature of the third dimension, though. It's simply a trait of the flatlander. However, it's worth considering that we regularly observe objects which are larger than a point. This suggests that 'motion' through space independent of time may simply be so common that we don't notice it. If we consider the fly as a point then it's not apparent, but if we remember that the fly is a collection of particles then even when t is the delta the fly is occupying more than one 'point'.
Talking about the fly being in two spots at once was a little sloppy of me; the more precise thing to say would have been that the fly's movement could satisfy things like dx/dt = 0 or dx/dy = 0, but never things like dt/dx = 0, with dt in the numerator. According to special relativity, that much is true regardless of how the fly interacts with the fourth dimension.
As you say, you can view the fly as a collection of particles; your suggestion, then, is that there's one 'fly' particle, and at any particular time-step it moves through the positions of all the elementary particles in the fly? Even disregarding the issue of multiple elementary particle types, that's not movement in the sense that I was discussing - it's a jump, not a smooth progression.