Proving that $0\cdot x=0$ using field axioms
You can't.
Let $F=\Bbb Q$, define addition as usual and $$x\cdot y =\begin{cases}xy&\text{if }x\ne 0\\y&\text{if }x=0\end{cases}$$ Then
- $(F,+)$ is an abelian group because $\Bbb Q$ really is a field;
- $(F\setminus\{0\},\cdot)$ is an abelian group because $\Bbb Q$ really is a field and $\cdot $ conincides with standard multiplication here
- Left distribution holds for $a\ne 0$ because it holds in the field $\Bbb Q$
- left distribution holds for $a=0$ by direct verification
In other words: your collection of axioms is "wrong".