facts on tree and MST
Assumption $II$ is true for any general graph with non-negative edge weights, including the graph $G$ with specific properties. To understand this, think about the inverse of the statement.
Let's assume the weight a $u,v$ edge is lighter than edge $e$ on the shortest path. The shortest path is made out of $e$ + $C$ where $C$ is some constant. Because $C &gt;= 0$(non-negative graph) and $e$ &lt; edge between $u,v$, by definition the shortest path we have assumed to be optimal is not(because the edge directly connecting $u$ to $v$ would be shorter).
To your specific graph into this, this lighter edge from $u,v$ cannot exist because if it did, it would have to be part of the minimum spanning tree(and thus part of the shortest path as mentioned in your description).