Unless you annex them, no, they don't. You don't need them to build the requirements for national wonders either. It's one of the reasons warmongering is still the best path to follow right now: you get nearly all the benefits from extra cities, while not suffering some of the penalties (only penalty is the happiness, and maybe the diplo hit).
A city being razed counts as annexed while it exists; you can stop razing it if you want, and start again later. The only thing you can't do is pupet it once annexed.
EDIT: ninja'd.