The real worry is when you're caught in a war while suffering extreme unhappiness. Its something you never ever want to be in at all costs. Defeating enemy units become very difficult and forces you to use alot of tactics to overcome the lemming rush trying to crush your now weak units.
You might survive or it might send you into a death spiral.
It happened to me several times but I survived due to the fact I'm fighting the AI not human players. lol
I'm saying that it is only manageable if you're suffering extreme unhappiness DUE to being in a war and capturing cities. As long as you protect your weak units such as artillery/ranged from mounted/armor units, you're fine. You can also solve the unhappiness issue by selling your razing cities to another AI. Either that AI will liberate, puppet or raze that city. Liberate is the worst possible outcome but can be avoided if you choose carefully who to sell to. Usually I can't even get back into positive happiness until capturing Forbidden Palace and/or passing world ideology to offset the ideology pressure. A good example is the recent CDG #4 as Spain where one of the achievement is to raze ALL non capital cities on the map. I frequently dipped below -20, and at its peak I was razing 14 cities at the same time, and wished cities would flip so that I can capture them again with much reduced pop, hence reduced unhappiness.