Showing that a dense subspace $Y$ of a first countable separable topological space is separable

This proof looks fine. Quite detailed. See Daniel's comment for an alternative faster proof.

To see that you need the first countable assumption on $X$: if $X=[0,1]^\mathbb{R}$, then $X$ is separable (but not first countable), and $Y=\Sigma_0[0,1]^\mathbb{R} := |\{f \in X: |\{x: f(x) \neq 0\}| \le \aleph_0 \}$ is dense in $X$ and not separable. Think about it.