Will MST find the shortest path for each pair $(r,v)$?
No, the MST may not find the shortest path for each pair of nodes $(r,v)$.

For example, consider a graph of three vertices $A,B,C$ with edge weight $AB\mapsto 2, AC\mapsto 2, BC\mapsto 3$. Its MST is edges AB and AC. However, the shortest path between $B$ and $C$ is the edge BC.