Uh, there MAY be an event for japanese surrender. In any case, either accept that or annex.
Bombs are produced automatically once you have lvl 6 reactors and lvl 6 nuke tech (which you do have). Also make sure that you built nuclear reactors, not power plants. In any case, the nukes are counted at the top of the screen, at the nuke sign. Hover over the sign to see when the next nuke is available.
Also, if you want to avert WW3, unally with the koreans. That's usually the main trigger you cant control.
That's normal - only the leader of the alliance can expel a member. Now that you're alone with your puppets, you can release them as fully independent states that will be allied to you - and then you can expel them from the alliance.