Let $V$ be a normed space and $x, y\in V$ with $x\neq y$. Prove that there exists a $g\in V^*$ so that $g(x)\neq g(y)$.
My proof is that apply the following Lemma
Let $V\neq \{0\}$ be a normed space. For $x\in V$, there exists $g\in V^*$ so that $\|g\|=1$ and $g(x)=\|x\|$.
By this Lemma, for $x-y\in V$, there exists $g\in V^*$ so that $$ g(x-y)=\|x-y\| $$ So $x\neq y$, then $g(x-y)=\|x-y\|\neq 0$. Hence, $g(x)\neq g(y)$.
Does this proof work?
Moreover, I would like if we can apply the following result to get another proof?
Let $V$ be a normed space and $Y$ be a closed subspace of $V$. For $x\notin Y$, there eixsts $F\in V^*$ so that $$ F|_Y=0, F(x)=1 $$ and $$ \|F\|=\frac{1}{dist(x, Y)} $$
Your argument is correct.
The second argument would be to let $Y=\operatorname{span}\{y\}$. Then by the result you mention you would get $F\in V^*$ such that $F(y)=0$ and $F(x)=1$.
Both results are good examples of the power of the Hahn-Banach Theorem.