Differences between Minimum Spanning Tree and Shortest Path Tree

AD a)

a very simple visualization would be:

MSP in a graph:

Shortest Path between A and C:

AD b)

I guess that the uniqueness of MSP means that there is only 1 MSP MSP in a graph. So: First) Yes, if the weights of edges are not distinct, multiple MST may exist at the same time. In case of our graph, another possible MSP would include arc D-C instead of A-B (as an example). Second) The uniqueness of MST does not influence the answer for a). As an example:

So lets take a look at a very simple graph:

(A)---2----(B)----2---(C)
 \                    /
  ---------3----------

The minimum spanning tree for this graph consists of the two edges A-B and B-C. No other set of edges form a minimum spanning tree.

But of course, the shortest path from A to C is A-C, which does not exist in the MST.

EDIT

So to answer part (b) the answer is no, because there is a shorter path that exists that is not in the MST.

Regarding (a), I agree.

Regarding (b), for some graphs, there may be more minimal spanning trees with the same weight. Consider a circle C3 with vertices a,b,c; weights a->b=1, b->c=2, a->c=2. This graph has two MSTs, {a-b-c} and {c-a-b}.

Nevertheless, your counterexample still holds, because the MST is unique there.