Understanding the hint of a question to show that $G$ is Abelian.

Here is the question I want to answer:

Let $G$ be a finite group such that 3 does not divide $|G|$ and such that the identity $(xy)^3 = x^3 y^3$ holds for all $x,y in G.$ Show that $G$ is abelian.

And here is the hint I got for the question:

First show that the map $G rightarrow G$ given by $x mapsto x^3$ is bijective. Then show that $x^2 in Z(G)$ for all $x in G.$

My questions are:

1- Could anyone explain for me why we should show that the map $G rightarrow G$ given by $x mapsto x^3$ is bijective?

2- Also, why we should show that $x^2 in Z(G)$ for all $x in G$?

3- And how proving the hint will show that $G$ is Abelian?