“The only thing that I do not understand is, why Hiroshima and Nagasaki? Why not Tokyo, centered on the “false emperor’s” palace?”
The specific answers to those questions are: Hiro and Naga had not been bombed, as had many Japanese cities. They were selected because the military wanted to see the destructive effects of the bombs without prior conventional bomb damage. Tokyo in fact had itself been firebombed in an incendiary attack that killed an estimated 100K and flattened many square miles of the city. Additionally, there was quite a bit of sensitivity as to how destroying the religious foundation of Japan might well be counterproductive....for whatever reasons you may imagine. Tokyo, yes, was the capital. But the spiritual center of Japan was Kyoto, which was spared a lot of bombing that it otherwise might have been subjected to.
I don’t understand your “islands” question. If you mean “we should have demo’ed the bomb”. That was thought of. At the time, we did not have dozens of them, I think there were ten or so. These things were thought about in great depth, and remember that most of the Manhattan scientists were relatively liberal and did not wish to destroy Japan, only to punish them and bring the war to the most expeditious close possible. There was no real question that Japan would eventually lose the war, and while some of course wanted to inflict massive punishment, the decisions were made after tremendous debate and consternation. I believe they were the right ones, others may disagree. They unquestionably had precisely the intended effect.
Please see my post # 23.
There were three. One was detonated on the top of the test tower.