The expectation of $S$ is indeed $0$. This follows by the optional stopping theorem; see e.g. THM 29.11 with $X=W$, $S=0$, and $T:=\inf\{t\ge0\colon W_t\notin(-1,0.1)\}$.

The expectation of $S$ is indeed $0$. This follows by the optional stopping theorem; see e.g. THM 29.11 with $X=W$, $S=0$, and $T:=\inf\{t\ge0\colon W_t\notin(-1,0.1)\}$.

Alternatively and more specifically, one can use Wald's lemma for the Brownian motion -- see e.g. THM 29.12 in the linked paper.