Very interesting post. My 2 cents:
Why maths can be applied? Whatever one's approach to understanding our universe is, we have to agree that, as it exists, there needs to be some notion of "inner coherence" (call it physical laws or whatever). On the other hand there is maths which in my understanding is no more (no less) than a grammar whose associated language is all that is coherent (I should extend here but will not). To me, physics study this particular "coherent reality" and maths study any possible "coherent reality". N.B. a topological variety is a potential reality to me as long as it is coherent and can be described "grammatically", so to speak.