Typically, if you are going to starch, you want to starch before cutting. Starch stabilizes fabric so that it does not as easily stretch or distort while handling. This means it helps while cutting, and also while piecing.
The only thing about starching pre-cuts is that they might shrink. Shrinkage is not a problem once a top is quilted to batting and backing. However, if your pre-cuts shrinks before you use them, they might shrink too much for your piecing project. For example, I would hesitate to starch 5" squares if I were using a pattern that called for exact 5" squares; they might shrink in one direction or another to less than 5". Same with fat quarters, although usually even if the fat quarter shrinks a little you will still be able to get the required pieces out of it.
Usually, if I am going to starch, I am starching yardage. Storing starched yardage is a problem probably only in climates where there are a lot of bugs, especially if there is dampness and heat. Honestly, it's never been a problem for me in MN and WI.