Can a graph with more than |V| - 1 edges have the maximum weighted edge as part of its MST?
Consider the following example:

You can clearly see that the heaviest edge (connecting vertices 3 and 4) is in any MST of this graph. In general, the heaviest edge (assuming it is unique) will be part of the MST of a graph if and only if it is a bridge (meaning that if we remove that edge the graph gets disconnected). The condition $|E| \geq |V| - 1$ doesn't really change anything. However, in order for MST to make sense, we need the graph to be connected. Note that $|E| \geq |V| - 1$ does not guarantee that.