And does it matter, in choosing the nap, what the sheen of paint will be used over the primer? E.g. if I do my bathroom in semi-gloss using 3/8" nap vs living room in flat with 1/2" nap, should I also prime the bathroom with 3/8" nap and the living room with 1/2" or it doesn't matter?
Generally the smoother the finish the smaller the nap. The nap on the primer wouldn't matter in this situation. I doubt your walls were done with perfection - no offense (none of mine are either). So you need a certain amount of primer to get to the point of painting. Meaning if you use a smaller nap for primer you will just have to make more swipes to get to the same point. Most painters I know won't prime with anything smaller than a 1/2 inch.