I do not believe he would declare war on North Korea. They currently don't pose a major threat. As a bonus, a war with NK might anger their strong backer, China, who would not want a U.S allied country on their doorstep. As an added hinderance, Russia might also find their Pacific interests in the Vladivostok Port hindered by an American NK.
Apart from a nuclear launch or two. Missile defences aren't perfect, and the domestic and international outcry would be massive. Economic pressure from China may cause problems given the US's dependence on their industry.
America could counter any nuclear launch that comes their way. However, I would agree that Trump wouldn't do such a thing at this point because he is working to make us more independent from China's... coddling.
The American missile defence systems aren't perfect. It's like hitting a bullet with another one. A single missile isn't too difficult with the sophisticated systems we have now, but a thousand Russian missiles is a different story.
Secondly, China is coddling?
Yeah that's a terrible term for it. But for many Americans that's what it seems like because we literally pay China to get rid of our garbage for us.
Secondly, it seems unlikely Russia would meet us with 1,000 missiles coming our way, but in the event that it does happen, America is not totally hopeless. https://video.foxnews.com/v/5502750399001/
The THAAD system is a valid point, but it covers short to medium range missiles. The ICBM's are met by a different line of defence. However, countries know that nuclear war is the last thing anyone wants.
A conventional war may degenerate into a proxy war with China, which would drain resources and cause economic trouble for all parties.