Efficiently compute the sum of edge weights product of all possible path in a DAG
You can actually solve this in $O(|V|+|E|)$ time. Since it is a DAG, we can apply dynamic programming.

Let $p = u_1u_2\dots u_k$ be a path and $cost(p) = w(u_1,u_2)\times w(u_2,u_3)\times\dots \times w(u_{k-1}, u_k)$ be the edge weight product of the path. One important observation is that $cost(u_0p_1) = w(u_0,u_1)\times cost(p_1)$.

Now, for each node $u$, let $dp(u)$ be the sum of edge weight products of all paths originating from $u$.

Let the children of $u$ be $v_1,v_2,\dots,v_m$. We can compute $dp(u)$ using the relation

$$dp(u)=\sum_{i=0}^m w(u,v_i) \times dp(v_i)$$

A Topological sorting can find the order of nodes to compute. Then, the final answer is the sum of all $dp$

$$\sum_{u\in V} dp(u)$$