Correct answer only !

during which part of the basketball's bounce is the most kinetic energy gained?

From B to C. The maximum potential energy is equal to the maximum kinetic energy, meaning that where the object is falling from the max height the maximum kinetic energy will be gained. Since its obvious that the basketball bounces less in its second bounce we can get rid of all the answers having to do with it. From A to B, the ball has all kinetic energy initially that is then turning into gravitational energy as it is moving up higher. On its way down or from B to C, the ball is gaining KE as it is moving from a certain height downwards and since energy is conserved, the PE that it gains must be equal to the KE it lost or the other way around. So, from B to C the ball will be gaining the most KE.