Self-promotion, but I would like to mention my question  It contains two questions.  I have resolved Q1 and posted it as an answer (hence the question is no longer "unanswered") but Q2 has not been resolved.  I quote it here for visibility.

> **Q2.** Let $X$ be a Banach space.  Let us say a linear subspace $E \subset X$ **determines weak-* convergence** (of sequences) if for every sequence $\{f_n\} \subset X^*$ such that $f_n(x) \to 0$ for every $x \in E$, we have $f_n(x) \to 0$ for every $x \in X$.  Is it true that every such $E$ is nonmeager? 

The converse is an easy exercise with the uniform boundedness principle.