If the USA attacks another NATO country, article 5 requires that the rest of NATO all pile on. None of those countries could individually do much, but you'd be loosing hundreds of thousands of soldiers and trillions of dollars to possibly take an icy rock with a population of 56000. It's fucking stupid.
In one fell swoop the US would be dismembering itself from its most important geopolitical allies and secondary trade partners. There is absolutely no win here for the US unless NATO balks at US intimidation and decides to let Greenland go without a military response, which would open the door for more US bullying for lunch money. NATO must resist with force. Greenland is not worth the consequences, no matter how many minerals or shipping routes control there are in it. I suspect Trump will try the intimidation to the last moment possible, but i have big doubts that he will actually invade or be allowed to invade and intimidation never worked well against Europeans.
Depends on how the war goes. Nobody uses nuclear weapons as a first response but they're always there as a reminder that if my country goes down, so does yours.
Besides, intercontinental force projection becomes a lot more complicated when your opponent has the ability, theoretical or not, to sink entire carrier groups at once.
I think it'd be more likely that Trump would get immediately kicked out of office if he actually managed to start a war with the EU. There's a limit to how stupid the GOP is willing to be for him.
Counterproductive policies that can be used to pocket some more money? Sure, why not. But a war that threatens to fuck the economy, dramatically curtail international trade, and probably hand global hegemony over to China? Too risky for too small a reward.