Proving that $S=\{ x \in (X,\| \cdot \|) : \|x\| =1 \}$ is a closed set.

You cannot define freely the complement of a set. In this case,$$S^\complement=\bigl\{x\in X\,|\,\|x\|>1\bigr\}\cup\bigl\{x\in X\,|\,\|x\|<1\bigr\}.$$Yes, we can prove that it is an open set, but it is simpler to see that $S$ is the inverse image of the closed set $\{1\}$ with respect to the continuous function $\|\cdot\|$. It follows that $S$ is closed.