Subjects like nuclear armageddon are simply too important to be taken seriously.
Yes
Dr Strangelove could be backstory for a post-apocalyptic society.
It would work for anything from A Canticle for Leibowitz to the Fallout games.
I agree. It’s a pre-apocalyptic movie that gets as close as possible to crossing that boundary, without doing so.
Though I would argue that there’s not a hard cutoff between the pre-apocalypse and post-apocalypse. There is also a period where the apocalypse is actively occurring. So there’s a pretty big barrier to it becoming post-apocalyptic.
No, because it’s about how the apocalypse happens, not what people do after.
There’s also nothing sci-fi about it. The Russians could have built a doomsday device, they just didn’t (as far as I know).
It’s not “post” apocalyptic, so no. I’m not sure I’d call it sci-fi either, but it’s more applicable.