Accordingly, the Language of Set Theory (in this case using $ZF$ axioms) is built up with the aim to express all mathematics. Now, I know that, for example, the construction of the numbers ($\mathbf{ \mathbb{N},\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}}$) is got right from the axioms and definitions from the Language of Set Theory, so everything is inside the same theory. But, when we are dealing with especial structures like for example Euclidian Geometry, we need to state new axioms, and, apparently, it doesn't make any sense to add this axioms to our original set theory because it might be possible to find a different model, maybe completly artificial, in which some axiom of Euclidian geometry migth not hold or result in a contradiction. But Euclidian Geometry and all the different theories that one might think of are part of mathematics. So my point is that it is impossible to develop all of mathematics just from the axioms of set theory. Does it make sense? Please tell me if I'm wrong and if I'm missing something. Also I have these questions:
In building these new theories (like euclidean geometrie, etc), each theory uses the language of set theory. But also we add new axioms. So, are we talking of a different language? Are we talking of a different theory? Is the language of set theory a metalanguage in this new language? I'm totally confused.