I would offer a resounding "No!" to those situations.
Let's look at them separately.
First, take the idea of a US first strike. Should a president -- and go ahead and think in terms of our current mentally unbalanced President Trump as you ponder this question -- be able on his own with no input from the Senate to decide to launch nuclear-tipped missiles at another country that is not attacking us, simply to deter them from doing so?
Some might say, well these days most wars and invasions the US has launched have been set in motion by presidents on their own. The last time Congress was asked to weigh in at all was in 2003, and even then the debate was just pro forma, since the Bush/Cheney administration had already moved US forces into position to invade Iraq, and would probably have gone ahead and invaded even if Congress had voted "no." But in any case, nuclear war is different. It is likely to quickly expand beyond existing battlefields and countries. China, for example, has noted pointedly that it has a defense treaty with North Korea which states that if North Korea is attacked, China will come to its defense (as it did during the Korean War). So deciding to launch nukes against North Korea could mean triggering a Chinese nuclear attack on the US. And because of the nature of nuclear weapons and how they are delivered, such decisions and such a spreading of the conflict would all be happening in minutes, not days or weeks. Indeed a president considering an attack North Korea might well decide to attack China simultaneously with North Korea, knowing that if he didn't he'd be risking having the US attacked first by China.
Clearly this should not be allowed to happen. A president should not have the authority to launch a pre-emptive nuclear strike under any circumstances. (Indeed presidents should not be able to initiate wars on their own, nuclear or not. Whatever facile legal hacks like former G W Bush Justice Department attorney John Yoo might argue, that was clearly the position of the authors of the US Constitution, which states that only Congress may declare war. But that's a subject for another article.)
But, you might ask, what if satellite images showed some country like North Korea preparing to launch several large presumably nuclear-tipped ICBMs towards the US? Shouldn't a president then have the right to strike first to prevent such an attack?
I would argue no. It would be overkill and could, as mentioned above, trigger a nuclear attack from China on the US. The president would have the right and duty, if convinced that an attack on the US was imminent (and that means imminent, not possible in a month or two or a year down the road), to destroy the threat, but the US has that capability without relying on nuclear weapons. Missiles may be dangerous weapons and hard to knock down once launched, but on the ground they are delicate machines, easily disabled or destroyed. A president has the authority to order such an action without going nuclear. (In fact, unlike the US Iraq invasion or Libyan invasion or this country's current military action in Syria -- al violations of international law -- such an attack would be legal under the terms of the UN Charter, which permits the initiation of a war if a country feels an attack on itself is imminent.)
I should note here that while Trump and the majority of the US media have been portraying North Korea's leader, Kim Jong-un, as a crazy person who might commit national suicide by lobbing a missile or two at the US, or at the US military base on Guam, to prove some insane point, experts who know North Korea and understand its history say Kim is not crazy at all, and has no intention or desire to destroy his country and himself. Rather, he has recognized that nuclear nations don't get invaded by the US, and sees having a demonstrably credible nuclear weapon and delivery system as a way to finally have his country recognized and to force the US to finally agree to negotiate a formal end to the Korean War.
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