Max-Flow: Detect if a given edge is found in some Min-Cut
Here is a solution for the first question: Suppose $w(e)$ is the weight of $e$, calculate min-cut value for $G$, suppose is $C$. Then we remove $e$ from $G$ to make $G'$; again we calculate the min-cut value for $G'$, suppose is $C'$, if $C-C'\ge w(e)$, then this concludes that $e$, participating in at least one min-cut (that you already know it), otherwise $e$ does not belong to any min-cut. 

Link to SO answer.