Hey Conrad,
Thanks for checking out my paper and bringing up these points. I'll check yours out as well, but for the moment I'll try to comment on your observations.
I didn't address the relative importance of certain axioms for exactly the reason you identify - the diversity of meanings makes the discussion quite complex. A single axiom might even have different importance in different subfields of physics ("atoms are molecules" is a good example). But I don't think the existence of these different meanings has any real bearing on my basic thesis.
* I'm implicitly using the idea that we are not modeling the "real" universe, but rather "the observable" universe. So, the spin of an electron may be a very important quantity in the model, but the importance of that axiom to the underlying pattern of nature is unclear - or even nonexistent. So we wouldn't have QED without the internal symmetry properties of the electron, but we wouldn't have plane geometry without the parallel postulate. It seems to me they have similar roles here.
* I'll just quickly comment that there are certainly mathematical axioms which are of differing importance - at the moment, the Axiom of Choice comes to mind. Without it, some basic topological results are not true (product of compact spaces are compact), but with it, you can get some strange results (Banach-Tarski, for instance).
* As for the interactions that you bring up, I think it's fair to say interactions in mathematics are more diverse than in physics. Interactions in physics are essentially governed by partial differential equations, so they lie in realm of Analysis. There are topological interactions as well, which also belong to mathematics. But how the axioms of Euclid (lines and points) interact to give us theorems is certainly different than how we use the interaction of particles to give us predictions - essentially just applied calculus. So *why* these axioms exist come from outside mathematics - from nature - but with the axiom of measurement you can bring them into the formal system.
Again, thanks very much for reading and commenting. Your thoughts are worth further consideration to be sure, but my first instinct is that since I'm not actually assigning importance to the abstraction beyond the observable universe, the internal symmetry of an electron has no more significance than something like the parallel postulate.
Chris