If G has a unique minimum spanning tree, does that mean the edge weights in G are also unique? if yes why and if no why?
The question is: If $G$ has a unique MST, then are the edge weights necessarily distinct?
The answer is no. If $G$ is a tree with all edges having the same weight, then $G$ has a unique MST.
More generally, a minimum weight spanning tree must contain every edge that is a bridge (a bridge is an edge whose removal disconnects the graph). Consider a graph which contains two bridges, both having the same weight, and with the edges in the rest of the graph having distinct weights. Such a graph has a unique MST even though the edge weights are not all distinct because the bridges have the same weight.