Many times there are problems which are, in a sense "outside of geometry", but are nevertheless amenable to a geometric approach. For example, I may be asked to prove that the ranges of any two affine one dimensional functions must have either one, zero, or infinitely many values in common. An easy proof is to say, "One-dimensional affine functions are lines in Euclidean geometry. Such lines intersect once, no times, or infinitely many times according to Euclid's postulates."
In another case, we may be asked to prove a trigonometric identity or inequality. But instead of appealing to their analytic definitions, we may often draw a circle and some lines and use the theorems of Euclidean geometry.
The theories of linear equations and trigonometry can be developed without bringing in any axioms of Euclid. Typically both theories can be derived from the axioms of set theory. The postulates of Euclid, however, cannot be so derived.
1) Why is Euclidean geometry special in this way? Why won't we be able to use another sort of geometry (hyperbolic?) to such proofs? In other words, Euclidean geometry seems more natural than other sorts - but why?
2) Are these proofs rigorous? Should a geometric argument be just as convincing as a proof in language of the original theory?
You don't need Euclidean geometry to prove that lines intersect once, no times, or infinitely many times. This can be proven in affine geometry, the point being that affine geometry has a notion of lines and parallel lines but not a notion of length. Affine geometry in turn is more or less linear algebra, which is thoroughly embedded into modern mathematics.
Once you prove that linear algebra in $\mathbb{R}^n$ (possibly as an inner product space if you need lengths or angles) models whatever part of Euclidean geometry you need, a proof using Euclidean geometry is completely rigorous. Other kinds of geometry are also useful in mathematics (e.g. hyperbolic geometry is indispensable to low-dimensional topology) but the basic objects in them (e.g. lines) are not as straightforward (!) as lines in Euclidean geometry.
I can't really make sense of the statement
Would you say that the axioms of group theory can't be derived from the axioms of set theory? In my opinion this is a "wrong question." You can write down Euclid's axioms and ask what sorts of models they have, and you can certainly write down models of Euclid's axioms using ordinary set theory.