I have found the "spread culture" mission to be virtually useless. Each such mission, when successful, adds 5% of whatever culture you already have in the city, so you must already have some culture there: the target city thus has to be close to one of your existing cities, preferably one with high culture. Also each mission costs more EPs than the previous one: you may lose your spy: successful spies go back to your capital and hence need time to go back and try again, during which time the enemy city is still pumping out its own culture faster than you can make yours grow.
By my calculation, 12 such succesful missions should put your culture up from 17% (as in post #12) to 30.5%, which won't really do you any good.
A massive sabotage raid before invasion, on the other hand, is definitely good. I like to wipe out the enemy resource developments, sending their cities into unhealthiness and misery and thereby hopefully losing population, thus reducing the time taken for resistance to cease after capture. Sabotaging such developments close to your border can be done a turn or two before you attack: the AI will send Workers to redevelop, easy pickings when your troops move in. Causing city revolts is of course best done only in the cities you're going to attack in the same turn: such revolts are quite expensive, and unnecessary if you have enough siege weapons or bombers