Second question: Axioms are statements assumed to be true for the purposes of a theorem. If they're later proven wrong, then you have a theorem that's correct but doesn't reflect reality. Some of them end up being necessary - for example, there's no mathematical way to prove that two parallel lines don't intersect or that triangles have 180 degrees, despite the world working that way.
Nope. Since frostshotgg pointed out the problem with the definition, I'll simply add that changing your axioms can create all sorts of interesting world-views that do pertain to reality. For example, what if I told you that two parallel lines can in fact intersect? Euclid's fifth is an awkward, cumbersome, and actually unnecessary axiom. Its use defines realm of Euclidian space, but discarding it and making either one of two assumptions - either that parallel lines always intersect or that they always diverge (get further away from each other) creates all sorts of interesting and new realms of mathematical theory, some with practical applications. While mostly famous for its use as a term in Lovecraft's works, the most famous non-fictional example of non-Euclidean geometry is actually one everyone is familiar with: the Earth's surface. Lines of longitude are parallel lines that intersect at precisely two points (the north and south poles), and triangles do not sum to 180 degrees (for example, draw a triangle using two longitudinal lines and the equator). Any curved space is non-Euclidian in nature - a two-dimensional sheet with positive curvature (elliptic) describes, at its simplest, a sphere, and with negative curvature (hyperbolic) a saddle - and this has seen great use in understanding space and gravity.
EDIT:
Oh, I must also hasten to add that this does not mean that Euclidean space is worthless, any more than general relativity made Newton worthless. It's still all sorts of useful for what it does describe - flat space, or more generally space where the local curvature approaches zero. When dealing with problems on the scale of meters, it's not as important to worry about a radius of curvature measured in thousands of kilometers.