Attempt at solution
This is an old exam question. Am I correct to say the shortest path goes through M since M is in d(O,M) and d(D,M)?
I don't know how to prove this and answer the question about the cost
With some assumption about the search algorithm (i. e. one that does not miss nodes with smaller costs - check this), you are right.
The fact that there are no shorter paths looks trivial then.