How hard is it to convert, say, a 20 cc weed whip engine into an engine for an aircraft. I know I will have to shave weight, but what else is involved?

The hard part is knowing what engine to start with. The fun part is doing the work. Places like CH have bolt on parts to ease the process. After what I've learned from doing a bunch of them I'll never buy a engine less than 50cc again. There's some that are just plain easy to do and produce excellent engines.
As a place to start the Homelite 25/30cc engines are about the easiest to do and will be the least vibrating engines with acceptable power.