A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.
My question is this: Is there a first countable and star countable space which is not separable?
Example 2.20 of this paper by Petra Staynova is an example, due to R. Winkler, of a first countable, Lindelöf (and hence star-countable), Hausdorff space that is ccc but not separable. Winkler’s paper can be found here.
$[0,1]\times[0,1]$ with the lexicographic order topology is compact and first countable but not separable.
Added: I almost forgot: $\mathsf{CH}$ implies that there is a first countable $L$-space, i.e. a first countable space that is $T_3$, hereditarily Lindelöf, and non-separable, while $\mathsf{MA}+\neg\mathsf{CH}$ implies that no such space exists.