Answer to Q1:
Let $(X\ d)$ be a metric space. I call $A\subseteq X$ $\epsilon$-dispersed $\quad\Leftarrow:\Rightarrow\quad\forall_{x\ y\in X}\ \left(\left(x\ne y\right)\Rightarrow d(x\ y)\ge \epsilon\right)$.
Let $A_\epsilon$ be a maximal $\epsilon$-dispersed set in $(X\ d)$ for every $\epsilon > 0$ (apply Kuratowski-Zorn theorem). Then $\bigcup_{n=1}^\infty\ A_{\frac 1n}$ is dense in $(X\ d)$. (The rest is obvious).
Answer to Q2:
(I don't see any use for $\omega_1$--am I wrong?)
I call a topological space singular $\quad\Leftarrow:\Rightarrow\quad$ it has exactly one limit point (i.e. non-isolated).
THEOREM The topological product of an arbitrary Lindelöf space by an arbitrary singular Lindelöf space is Lindelöf.
PROOF In arbitrary singular Lindelöf space the complement of any open set, which contains the limit point, is countable. The rest is obvious.