Is the inverse of MST cut property true? Why? If we partition the nodes of a graph into sets A and B, there is an edge e of weight larger than any other edge crossing the cut between A and B, e would never be in the minimum spanning tree?
This is not the case.
If an edge $e$ is a bridge in the graph, then every MST has to include $e$. There can exist some cut in which $e$ is the heaviest edge, but this does not change the fact that $e$ must be included (there is some other cut in which $e$ is the only - and therefore the lighest - edge).