Consider a spanning tree of $G$. Then adding your edge $e$ creates a cycle. Removing any other edge of this cycle creates a spanning tree containing $e$. Hence there is no other edge in the cycle, which means that the edge must be a loop.
Part $(b)$ is only true if we assume that $G$ has multiple edges but no loops.
The Wikipedia page is a little misleading; it might be more accurate to say that at least one of the lowest weight edges will be in a minimum spanning tree (in a loopless graph).
This is apparent from the application of Kruskal's Algorithm for constructing a minimum spanning tree. Essentially, select the lowest weight vertex at each step that will not introduce a cycle. Because it is impossible to create a cycle when selecting the first edge in a loopless graph (per assumption) Kruskal's Algorithm will always select it.
In order to rule out the length $10$,$10$, $11$, and $13$ edges, they must all create cycles if they are included with the $8$ edge; this implies that there is a multiple edge with $8$,$10$,$10$,$11$, and $13$ all connecting the same two vertices.
I would say it's very non-standard to allow multiple-edges but not loops in the graph, because problems are generally a matter of simple vs. non-simple, where loops are allowed in non-simple graphs. Clearly, you can obtain a higher result by making $8$,$10$,$10$, and $11$ edges all loops, while using the other $4$ to create the spanning tree.
I would ask your professor to clarify this point, and definitely ask during an exam if a question is vague.
Best Answer
This is what the graph looks like in the $n-1=9$ case:
A spanning tree is a subgraph that is both (a) a tree and (b) contains every edge.
Here's an example of a spanning tree in the above graph:
There will be many others. The task set in the question is to find examples of spanning trees of diameter $d$ for $d \in \{2,3,\ldots,n-1\}$, for all $n \geq 1$.
Hint: The brown edges in the following diagram illustrate examples of spanning trees of diameter $2,\ldots,6$, respectively, in the case $n-1=6$.
I suggest you try to generalize this construction. (Note the diameter $n-2$ case is special.)