More benevolently, the AI makes a huge amount of money off of financial markets, uses the resources to start its own country, runs the country really, really well and expands citizenship to anyone who joins. Eventually, when the country is strong enough, the AI (with the genuine support of most people) uses military force to take over the world, giving us an AI monarchy.

Seems unlikely. Sure, it could be done, but it would waste a lot of time. I doubt a typical superintelligent agent would do that.

The AI identifies a group of extremely smart people and tricks them into answering the "hypothetical" question "how could an AI take over the world?"

I suspect this was meant as a joke, but while a superintelligent AI wouldn't need to do such a thing, a human looking for ways to destroy the world could use suggestions, so it might be a bad idea to give nonobvious suggestions that humans could implement.