Why does the USA demand that other nations are not permitted to research or produce nuclear weapons?

Only in your mind. Remember the story of the scorpion and the frog? He stung the frog while the frog was helping him across the river and they both drowned? The frog asked why and the scorpion said because "it's what I do." That's the way it is with Iran and Korea. As far as the bombing of Japan, you should read history. It was the only thing to do. You should know that.