You are asking big questions! My answer is brief but I hope it helps a little.
By axiomatizing a definition we can begin to prove theorems rigorously. If we formally abstract a concept and deduce formal statements from it (which can themselves be quite unituitive), we can be confident that the statements apply to the concrete situation at hand. On the other hand, if we always worked only with concrete or physical examples, we would have to figure out everything from scratch every time. A good general theory is one which concisely explains many specific cases at once.
For example: Can every periodic continuous function be approximated uniformly to whatever degree of precision we like by a partial Fourier sum? If we abstract away what's important, and study the situation in the general context of Hilbert spaces, we can answer this question not only for this specific case but also for a broad class of families of approximants in one fell swoop. We save a lot of effort by doing this, and we also usually gain extra insight into the problem.
Still, you don't want to abstract too early. It's important to understand the concrete case first before jumping a level in abstraction, otherwise you end up understanding nothing. If you're having any trouble with metric spaces (I'm not sure if you are) then you might find it helpful to look at a few specific examples to see what they're used for. Examples: In coding theory, we often use what's called the Hamming metric, the distance between two words. In graph theory, there is a natural geodesic metric: the shortest distance between two vertices as a walk along the edges. If that's not concrete enough, consider the popular "6 degrees of separation" rule: people are vertices and relationships are edges. Of course there are the usual examples of Euclidian space R^n, unitary space C^n, and the other various normed spaces you're studying in functional analysis.
By working with the axioms of metric spaces we can prove theorems which apply to all such cases at once. Here's an example of a tricky theorem: Let x be a point in a metric space. Suppose a subsequence satisfies the condition that each subsequence has a subsequence which converges to x. Then the entire sequence converges to x. You wouldn't want to prove this theorem from scratch in every specific concrete case! It's more efficient to abstract out the essential features (the axioms) and then prove the theorem in the general setting.
As for your other question: Set theory is (in my opinion) not a fundamental feature of mathematics. It just so happens that mathematicians today like to axiomatize everything in terms of set theory. Still, sets will always be useful even if we don't choose to define everything in terms of sets. A set is (loosely speaking) nothing more than a collection of objects, and we'll probably always find it useful to deal with collections of objects.
As for symbols: What more is a symbol than a hook on which we havg an abstraction? How can we manipulate abstractions if we don't have symbols for them? I'm using the word "symbol" in the broad sense here, which includes diagrams (or pieces of diagrams) and words in a natural language. If you want to see abstractions manipulated via diagrams and words, take a look at classical sources (say, pre-1500's). I believe our modern notation is far more amenable to manipulation and understanding.
In short: In mathematics, abstraction and symbol manipulation are the name of the game. If we have no symbols then we have no abstraction and no mathematics.