This is going to sound repetitive, but the answer is the field axioms, along with more basic axioms, such as substitution. Every single question you asked is answered by the field axioms or some other axioms. Opalg already proved that the only possible solution is -4.

I am assuming that we're working over the field $\displaystyle \mathbb{R}$. The field axioms aren't always relevant (our coefficients might not be from a field), but for now it's a reasonable assumption.