Centre, diameter, and radius of graph
A quick way of gaining some confidence or disproving a conjecture is using a computer to generate examples.

For Q1, if $x,y \in V$ such that $d(x,y) = D$, it is not true that any shortest path between $x$ and $y$ contains at least one central vertex (highlighted in red). Here, you can walk from 1 to 6 via 9, 5, and 8 avoiding the red vertex.



For Q2, the answer is also no. Below is a graph whose radius is 2 and diameter is 4. It has 3 central vertices (highlighted in red).



You can use ideas from these counterexamples to optimize the graphs further.