That's a bit too simple, not all functions work well as fields (differentiability is quite desirable) and you have no way to interact with the fields that way.
I think principal bundles come closest to what physicists call fields. Though I'm holding open the option that really the things in most equations are more like elements of the corresponding Lie-algebra.
It’s everything! Idk, I don’t think the universe owes us an answer there. What is a human? Well, it’s a human. You can think of all sorts of mental tools for understanding humans (eg “species”), but ultimately they just are.
Ok fair: they're things that right now are best understood using the term Field. I don't understand what kind of answer you're hoping for that would be better than this -- what kind of answer to "what is a particle" wouldn't be describable by mathematics?
By saying "the universe is fields", I'm saying "it's distributions of energy across spacetime". That's seemingly a consensus. Why demand that that energy must also form into strings or even tinier spheres or spheres in an alternate dimension or something? We have described fields in detail, I say Mission Accomplished
Let's say we describe a particle as being an excitation of a field. Can't we describe any physical phenomenon in the same way? For example, describe a football as an excitation of the football field. I think we all agree that this description does a terrible job of actually describing a football. In fact, I don't think it can be called a description at all. Likewise, I find describing a particle as an excitation of a field to be equally unsatisfactory.