Hardness of finding if a vertex lies on a simple directed path between two vertices
I just found a reduction from the two disjoint paths problem. Given a directed graph $G = (V, E)$ and two pairs $(s_1, t_1)$ and $(s_2, t_2)$, we can add a new vertex $w$ to the graph and an edge from $t_1$ to $w$ and from $w$ to $s_2$.

The new graph admits a simple path from $s_1$ to $t_2$ through $w$ if and only if the original graph admits two disjoint paths one from $s_1$ to $t_1$ and the other from $s_2$ to $t_2$.