The AI absolutely does cheat.
I'm struggling with the Aztecs on a sizeable continent, and finally have them on the run. Knocking off their last couple cities on the coast and they're extinct. I'm using my own settlers and razing the cities for slaves and building my own. Since mine are going in, the city area is smaller for these new cities until they build some culture. So, there's some open territory here and there between these cities.
What happens next? Other civs on other continents, who I have had no dialogue or seen before start racing ships toward this part of the map.
What a coincidence. A bit of land opens up and BANG, everyone's shipping settlers to it from across the globe.
Man, that's a pretty good advantage that the AI has - know what's going on all over the globe before they even mapped it (these civs just researched navigation, so they couldn't have gotten to this part of the map before).
Here's another well known one. Take a couple of cities boardering a civ. Go to war with them. Over defend one city and put only 1 defensive unit on the other and watch which city the civ sends it's stack of doom toward. Reload, but switch defensive units (or put that single defender on a back-row city) and see where their stack of doom goes.
Yet another advantage that would be nice to have as a player.