How to calculate $ \mathbb{E}\left[X|W=0\right] $
Solution 1:
$W=0$ iff $X \leq 0$. So $E(X|W=0)=E(X|X\leq 0)$. By symmetry of $N(0,1)$ we have $E(X|X>0)=E(-X|X <0)$. Also $E|X|=E(XI_{X>0})+E(XI_{X<0})$. Combine these to get $E(X|W=0)=\frac 1 2 E|X|$ and $E|X|=\sqrt {\frac 2 {\pi}}$