"While I disagree with the fact that humans are stupid, seeing as they are the smartest thing we know..."
Allow me to re-phrase what I meant, "Ignorant humans." All man is what we describe as smart. Some more than others, some less. The main threat is a consensus of ignorance. Ignorance of a fact, of a threat, of possibility and et cetra.
I would like to stress that these next parts consist of estimation, guesstimation, and theoretical possibilities.
And while nuclear war, in all accounts, is devastating, it would most likely not kill ALL humans. It would certainly hinder.. "Progress" with the human civilization, but nuclear exchange would most likely be limited, thanks to missile defense, mutually assured destruction, and the general thought that nuclear bombs hurt everyone. It also depends on what nuclear weapons hit where. An average missile of the DPRK can (in theory) hit anywhere in Japan. If it where to penetrate the defense systems the U.S. has in place in that region, and hit Tokyo, it would be devastating. BUT, their technological level in nuclear weapons is just above the bar of a dirty bomb. If my remembering is correct, they have somewhere between 6 and 9 kilotons of TNT in their latest nuclear test. In comparison, the bomb dropped on Hiroshima in WWII was 16kt of TNT. This, though tragic, would not cause as significant damage of the Little Boy. Not to mention the Japanese use concrete now instead of wood, and reinforce all multistory buildings for earthquakes. Bad? Very. But not threatening to global ecosystem or population, as the U.S. and all other retaliating forces would most likely not escalate the situation by responding with nuclear devices. But on to my next example.
Say the U.S., Russia and China all decided they hated each other (more so than usual) today. Russia, China and the U.S. begin full-scale nuclear exchange. Russian missiles, though historically flimsy (I have not seen any indication of upgraded delivery devices, though I am sure they exist), have quantity (an estimated 10,000), and powerful thermonuclear warheads that have the largest power range, up to 50mt (Though it is strongly suspected none above 30 have been manufactured, or are mountable on missiles). China has an estimated 240 nuclear weapons with high delivery quality, but slightly lower yield (Averaging somewhere between 660kt to 4mt). The U.S. has an estimated 10,000 missiles of high quality. These, however, have a moderate yield possibly peaking around 30, but having clusters of devices for any purpose.
If a nuclear exchange between these powers began, few would survive on the surface longer than 2-3 minutes after detonation in those nations population centers, and the centers of their allies. Smaller nations, such as Israel and Iran may take hits, and if possible, could respond with their own NBC weapons. That would lead to poisonous gasses, plant death, radiation, hyper pollution, sun blocking, ect. But humans could possibly survive. Some, not all, but just some. That would give hope that eventually they, if they survived, could re-inhabit the less radioactive damaged areas. This is because the majority of radiation after a nuclear blast of 10mt decays after two weeks. The next step is three months, in which surface living would be possible with respiratory devices. After 200-500 years, the vast majority of all radiation would be gone, and any nuclear winter would be fading or gone. This would lead to a "Nuclear Spring" where any plant life that may have survived, or could be replanted could possibly grow, and do well. After a specific amount of time, I'll give it 100 years for the lols, things would go downhill quickly. The blasting of the ozone, loss aerosols from the nuclear detonation and the settling of the 'dust', so to speak, would leave a slightly-much thinner ozone layer. This would lead to more water in the atmosphere, which could lead to more greenhouse gasses being secreted, which could lead to runaway climate change, turning us into Venus.
However, if the nuclear summer is short, does not occur, or is moderate, it can lead to an eventual restoration (I'm using that word EXTREMELY liberally) of the climate, which over time would lead to a restored biosphere.
But I am slightly off topic. My point being, human civilization /could/ live on after nuclear war. Not well or very happily, but it is possible.