I'm not sure how to do this (for instance, prologue would only have an any% category as you're only allowed to play as Snake and the other Task is during a rail shooter).

You can not, just list in the rules the category is not possible for those stages

On a similar note, is it possible for a run to be applied to multiple categories, or for one category to include another? Like, any Perfect Stealth No Kills run is also an S Rank run, and any S Rank run is also an any% run. Just so that the leaderboards automatically contain the best times for each category without having to duplicate runs.

I doubt it's possible. But 2 things for ILs:

People should always first summit in the highest standard category order is ANY%, ANY% S, Peacenik, all tasks then no traces (one day) and if i'm verifying a video that got a standard higher than the category it's summited in, i'm switching it to highest standard it can fit in.

If people want to be in all ladder (Any%, Any%S, peacenik), they wil have to summit in all ladders.