**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.