You're welcome for the details.

So I see 'they were ready to surrender' a lot in this thread, and while that's not... false, it's not exactly what it sounds like. They were ready to come to the table, yes, absolutely, but the problem was that they wanted to dictate their surrender - they wanted to keep their military, they wanted their industry rebuilt, they wanted the current government to stay in power- it was less of a surrender or more of a cessation of hostilities. Japan was 'ready to surrender' in much the same way Russia was 'ready to come to the peace table' about a year ago.

This was geopolitically not realistic, for a number of reasons- for one, allowing that kind of conditional surrender with Germany is directly what lead to WW2 in the first place, and nobody had any intentions of repeating that mistake. There was concern, given the view on surrendering, that it wouldn't actually be peace, or a surrender, merely a delaying tactic to build up forces and entrench. For another, Russia was bearing down on Japan, and the Allies wanted to limit Russia's geopolitical influence by preventing another East/West Germany. While the extra troops would have undoubtedly help save American lives, it would have ended in significant Russian and Japanese deaths, as well significant geopolitical issues long-term (East/West Germany worked so well, after all :P )

Long story short, the Allies absolutely wanted an unconditional surrender, exactly the kind of thing the Emperor and the military refused to contemplate, even after a single bomb was dropped. The military still refused to consider it even after the second, so seeing the a-bomb in action once would likely, I feel, not have done much.

RE: hitting civilians in large numbers, my understanding is less that they were deliberately targeting civilians, and more that they were looking for military targets that were geographically located in a position that would enhance the bomb's effects without considering civilians too much. You could argue in a very real way that they were deployed as terror weapons, or perhaps 'shock and awe' weapons if you want to be slightly less confrontational. Civilian casualties were, much like the entire rest of WW2, not much of a consideration- WW2 was considered a total war, and the Geneva Convention would not be signed for another 4 years, directly as a result of the atrocities of WW2. At the time, civilians were not considered something to inherently avoid unless you had some sort of political reason to do so (hence the leaflets). The most obvious example of this is the firebombings of Tokyo, which killed far, far more civilians in arguably far more painful ways, but there's plenty of example in the European front from all sides as well. Again, they were making decisions with the knowledge and viewpoints of the time. Doesn't excuse it, but trying to moralize decisions made in the past with current morals is always kind of a waste of time, in my opinion.

Regarding the third shot, there was, at the time, no bombs available when the uranium Little Boy bomb for Hiroshima was dropped, but they had prepped for another. They immediately turned towards trying to prepare another (Nagasaki's plutonium-based Fat Man), and managed to rush it to completion in just a week, but keep in mind that these were highly dangerous, experimental one-off prototypes being produced- it's why all of the planned subsequent bombs were of the fat man design, which was significantly safer, and America was completely out of uranium at that. It was only able to be rushed to completion so much because General Groves always planned to use two, and a lot of the logistics were already worked out and prepped beforehand. Before more plutonium bombs could be made, Woodrow Wilson called off the production. So yes, America was technically out of bombs, and completely out of uranium.

Arguably, America could have created more plutonium bombs, but was limited by the availability of plutonium (which is lengthy to turn into weapons grade), the speed at which they could be safely produced (and Fat Man was, frankly, very unsafely produced, it should have taken nearly 3 weeks to create), and America only had a small amount of weapons-grade plutonium stockpiled. So technically, both positions are correct- America only had two bombs, and they certainly could have made more, but they were limited by time and materials, and lack of willingness. They had, perhaps, one or two more fat mans they would be able to drop, with perhaps 3+ week production times for each (because no logistics were prepared for it), before it would have dropped to something like iirc 6 months per bomb due to lack of prepared plutonium.

So yes, one could argue there could have been more bombs after the first two, but it was generally considered by the American military and also the President that two was the 'magic number,' so there wasn't any setup for them, so they would not have been cranked out anywhere near as fast. Nobody believed that one bomb would trigger a surrender (because of, again, the cultural viewpoints on surrendering) as well the implicit belief that it would be a one-off prototype that could not be repeated.

If two did not, and it was widely considered it would, nobody believed 3 would be any more likely to trigger a surrender than two did, and might even convince them to fight harder. In addition, due to the effects of radiation, America would have limited to how they could use the bombs one the land invasion started- with Russia from the north, America from the south-east, and most of central Japan firebombed, there's not a lot of good targets without hitting allies.