Can we prove that axioms do not contradict? We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So:
Is it always possible to prove that a set of axioms does not contain a contradiction? (So if not could we perhaps find a contradiction in a structure that we definde by axioms someday?) Or is it proven that doing so is impossible?

For a first order theory, basically a set of "axioms" formulated in a first order language, you have Gödel's completeness theorem. The theorem establishes that a first order theory is consistent(i.e. non-contradictory) is and only if you have a model for that theory(i.e. a "real" mathematical object satisfying the axioms of the theory).
From this you can know that group theory, ring theory, field theory and, in general, every first order theory that has a model is a consistent theory.
To see this you proceed using the completeness theorem and observing, for example, that $(\mathbb{Z}_2,+)$ is a model for group theory, and $(\mathbb{Z}_2,+,.)$ is a model for field and ring theory.
