If $G$ is a semigroup (i.e. it is endowed with associative operation), why it is true that $x^{n+m}=x^nx^m$ for any $m,n \in \mathbb{N}$?
I fix $n \in \mathbb{N}$ and then use induction on $m$. If $m=0$, obvious. Suppose the claim true for $m$, what happens for $m+1$? May I write
$$ x^{n+m+1}=x^{n+m}x $$
and then continue by induction?
Yes, your proof is correct. The fact that $x^{N+1}=x^N x$ (along with $x^0 = 1$)is the definition of the exponent notation.