They dropped two nuclear bombs on Japan to make the Soviet Union's invasion of Hokkaido unnecessary and as a performative act to horrify the world in to subjugation.
The whole 'saved more lives than it cost' is a bad argument unless the plan was to slaughter a hundred thousand civilians on landing. Note the Nazi invasion of France cost 60,000 lives in civilian resistance...this was at the beginning of the war.
Japan would have surrendered on the first day of invasion.
This is accurate, to my knowledge. The firebombing killed more people than the nuclear bombs did, even.
The emperor and military were pretty far removed on whether to surrender. I don't have the sources on hand, but I believe the military was planning a coup before the emperor announced the surrender on national radio. The American projections on Japanese civilians fighting to the death or whatever was clearly wrong, but the military forcing the fight to continue until the capital was captured, same as Germany, is not that farfetched.