Let's review the spring question (Q6).
The question states that "[t]he ball and the spring come to rest when they are at a distance of 0.50 m below the uncompressed position of the spring."
The wording of "come to rest" is slightly ambiguous. It may mean they completely come to rest, or momentarily come to rest.
However using the most obvious definition of "at rest", we can assume this means when velocity is equal to zero, which would then mean when the ball and spring are first momentarily at rest (at the lowest point of spring compression).
Using this definition, then part a. can be done using a conservation of energy approach and the correct answer can be obtained (in fact the question can only be answered if you assume this definition).
Part b. asks for you to find the acceleration at maximum speed. Many people assumed that the maximum speed occurs just before the ball hits the spring. This is not true. It is true that the maximum acceleration occurs just before the ball hits the spring, however as the ball begins to compress the spring, the acceleration will decrease, during which time the velocity is still increasing, just at a slower rate.
Thus the maximum speed will then occur as acceleration decreases to zero.
For part c., you are asked to find the compression at maximum speed. From b., the acceleration at this point will be zero, and hence net force will be zero. Thus mg = kx, and solving for x (using k = 392 N/m given in part a.), x = 0.05 m.