is it usually required in more advanced math to memorize mensuration formulas such as sphere and cone volumes. etc?
Memorization is very useful in advanced mathematics. However beyond memorizing the formulas you need to know how to apply then, alter them to fit new situations, etc. I you memorize the basic formula like the area of a rectangle. width * length = area Then later you are given just an area and a width and have to find length. You must be able to adapt the memorized to fit the new data. So area/width=length. Without the understanding of basic algebra some people would be unable to solve this new problem. So make sure that you not only memorize put understand exactly what the formula does and its uses.
For the first the answer is no, you have to understand how to derive things if you need to, but when you need them you will look them up. For the second the answer yes, its a hoop you are made to jump through.