Showing that for some group it is abelian iff $x • (y • x ^{−1} ) = y$

This trivially follows from the following:

$$begin{align} x bullet (y bullet x^{-1}) = y &iff (x bullet y) bullet x^{-1}= y \ &iff ((x bullet y) bullet x^{-1})bullet x =y bullet x\ &iff (x bullet y) bullet (x^{-1} bullet x) = y bullet x\ &iff (x bullet y) bullet 1 = y bullet x\ &iff x bullet y = y bullet x. end{align}$$

Here I denoted the neutral element of the group with $1$. Try to understand why each equivalence holds (good exercise!).

No, it is not. All you did was to prove that if that condition holds, then the group is Abelian. But you also have to prove it in the other direction. That is not difficult, though: if the group is Abelian, then$$xbullet(ybullet x^{-1})=xbullet(x^{-1}bullet y)=(xbullet x^{-1})bullet y=y.$$