Why can mathematicians pick and choose axioms? Obviously, there need to be some 'self-evident' truths that we can't prove, but on which we base certain theorems; e.g. the axiom of choice lead to the well-ordering theorem (which could well be an axiom).
Essentially, what I'm asking is how do mathematicians know what is "so trivially true, that it is assumed to be true"?
For example, Playfair's axiom/the parallel postulate does not seem to be trivially true (at least to me, and from what I gather, many mathematicians tried to prove it), so why did was this axiomatised?
I understand that we can construct useful things from these unproved assumptions, like, for example, the reals, from the axiom of completeness, but, unless it is a defining property, how can one say for certain "statement  $X$ is unequivocally true"?
Finally, are there axioms yet to be discovered, and, if so, who determines their validity?
 A: Since by definition no axiom can be proved using only the other axioms, the choice of axioms is somewhat arbitrary as you can always decide to work within a system with less or more axioms. Mathematicians therefore choose axioms based on how useful the results based on those axioms can be. 
For instance, if we chose not to use the axiom of choice, we could not assume that a given vector space has a basis. Since a basis for a vector space is a very useful thing, assuming the axiom of choice whenever we deal with vector spaces is a common decision. You could still have a vector space without it, but it's properties would be far less useful than vector spaces axiomatized with  axiom of choice.
The same goes for the parallel postulate - thousands of years of mathematics used this as a basis for Euclidean geometry, trigonometry and so on. When a use case was found for non-Euclidean geometry, this postulate was removed from those use cases.
Ultimately, mathematics is all about creating structures that can be used - either to model real world problems, to further our understanding of existing structures, or to create new and interesting structures. The choice of axioms is just a means to an end, not a representation of some deeper truth.
A: A formalist perspective is that if something didn't satisfy the axioms of Euclidean geometry, then we wouldn't call it a Euclidean plane. If something doesn't satisfy Peano's axioms, then we won't call it a collection of natural numbers. And so forth.
A: Alternative idea of axiom: property verified by the structure we are interested in.
