MST: Is there such an example of a graph with unique mst and not unique light edge?
If there is not a unique light edge crossing any cut, it means that every node has at least 2 edges of minimum weight. If you use Prim's algorithm to build your MST which is:

Grow the tree, adding nodes one by one.
On each step, select the minimum weight
edge leading to a new node.


You realise that whatever is the starting node, you have at least 2 choices. If the MST was unique, there would be leaf nodes having only one possible edge. Therefore there are several possible MST.