Product of the entries of the product of two matrices
You are asking two questions. I will only answer the second one. Let $1$ be the all ones column vector of dimension $n$. The sum of all entries of $AB$ is $1^TAB1 = (1^TA)(B1)$, which gives an $O(n^2)$ algorithm: compute the vectors $1^TA$ and $B1$, each in $O(n^2)$, and then compute their inner product in $O(n)$. This corresponds to the identity
$$
\sum_{i,j,k=1}^n a_{ik} b_{kj} = \sum_{k=1}^n \left(\sum_{i=1}^n a_{ik}\right) \left(\sum_{j=1}^n b_{kj}\right).
$$

As for the product of all entries, I suspect you can't do better than the trivial algorithm which first computes the entire product matrix, and runs in time $O(n^\omega)$.