Mehran diagrammed all of the information from the stimulus in the top right corner. You are correct that he didn't do a contrapositive for each statement. When you are very comfortable with sufficient and necessary reasoning, you can see the contrapositives in your mind without writing them down. To me, when I see X - - - > Y, I am automatically thinking not Y - - - - - > not X. I won't always write it down, because it does take time.
However, you are welcome to use any strategy that will help you answer the question. It is ok to over-diagram as you practice your conditional reasoning. As you get better at it, you won't have to diagram quite as much and you will be faster. For now, do whatever it takes to understand the logic and answer the question correctly.