All my responses are gleaned from some peripheral searching on Copilot & the Googles as well as the book by Annie Jacobson, Nuclear Ware: A Scenario
1. I'm happy they illuminated the fact that we have a minimal amount of interceptors and they are notoriously inaccurate. It is literally like trying to hit a bullet with a bullet, just on a larger scale. To your point and the sure, possibility but our launch detection systems are, bar none (except maybe the aliens

) to the point we've offered it to the Russians in the past because their equipment is so bad. I expect there was a launch from NK.
2. I guess it could be but risking almost assured total destruction of your country seems like a big gamble to risk on a test. I'd suspect they would either blow up the ICBM if they had the capability or place a very timely call to China who would, in turn, call us.
3. Again, I guess but the US operates on a "fire on warning" footing which I think they briefly mentioned in the movie as if it was a discussion point. It is not and even if Mr. President doesn't pull the trigger, I suspect he is surrounded by enough hawks that it would happen regardless. What this means is if we suspect a nuclear launch from an aggressor nation, the US response is to pull up the 4-5 scenarios we have in place for that particular country, Mr. President picks one and we launch. The scenarios can range from taking out critical infrastructure to total annihilation. In the case of NK, there is a lot of consideration for collateral damage. You will most certainly take out a fair amount of South Korea and enough of China that they will not be happy.
4. The more likely scenario IMO. Sadly, when one nuke is launched, given the old technology and current launch on warning policy, my guess would be that we send a limited response back to NK. In the book I mentioned above, this is the exact scenario, NK launches a nuke at us. The response has to take place within a very short timeline, 30 minutes or less. The other issue with a NK response is, all of our weapons have to traverse through Russian airspace and if we don't or can't get a hold of the Russians and convince them that we really are blowing up NK, not them, then they for sure are going to empty silo's which then destroys the world.
In short, nothing good happens after the launch of a nuclear tipped ICBM and it happens in a very fast period of time. Within an hour, civilization as we know it, could end or at least be so drastically changed that we would be pretty close to starting from scratch.