[neutralino]

Interesting thread, MJA. Axioms are used mostly in pure mathematics, and can be thought of as strict rules which we must always follow. Whether a statement is true or false can then be derived, using logic, from the axioms. Thus, if there are no axioms, then we cannot say whether statements are true or false.

A conjecture is, as you say, something that one puts forward to be a true statement, but which has not yet been proven. However, conjectures are not just guesses. Normally, conjectures are made when we really think that something should be true, but just haven't been able to prove it yet.

I'm not sure that any of your questions are answerable. I don't think that we can ever, in nature, prove something completely, since the world is not an abstract mathematical framework. Any theory we have of the universe must be a simplification. We can prove things within the refines of a model, but whether this is absolute "truth" is debatable.