I am sorry, I should be off, so I will post the answers with explanations now.
Indeed the y-coordinate of the transition state determines the speed of the reaction! It is called the activation energy of the reaction, and you can view it as the barrier the molecules have to overcome to become products.
A catalyst speeds up a reaction not by changing anything about the reactants or the products, but by lowering this barrier. If you would add a catalyst and then draw the graph, the reactants and products would be at the same position, yet the peak would be much lower.
Now we compare the two graphs with the Hammond postulate in hand. The chlorination has the TS more or less in the middle, so it resembles product nor reactant. The fluorination has the TS more towards the reactants. We cannot compare the two barriers quantitatively, but intuitively, which barrier should be higher?