I was quite complacent about nuclear weapons after the end of the Cold War. I checked out targeting maps and all kinds of stuff when I was in university, seeing just what the plan was when the nuclear shit hit the population fan, and it was all very remote and now-we-look-back-on-follies-and-chuckle-dryly. Then I did some more reading and realised we're past the high point of nuclear tension, but nuclear armament is still very real, there's still plenty of it around and it could conceivably do the job to the extent the majority of us would care.
Anything else is, barring some sort of Holocaust for everyone not to the right of Hitler, peanuts. I've watched Threads, man. You can't scare me with conventional warfare. And I don't have kids to fear for.
Putin's Russia is officially Not Very Nice. They don't have the muscle to really be a global power, but they're still a serious force, and what we saw in the Crimea was nothing more than trying to bully the Ukraine into staying away from NATO and the EU, and to just allow itself to be dominated by Russia. It got out of hand, the Ukraine didn't fold, and the West backed them up. Their shenanigans in Syria are the perfect example of a kind of low-budget yet all-star performance: they get to test and display kit, they get to flex some muscle, they avoid taking lots of casualties, they get to talk a good game.
Putin doesn't want a proper fight, he wants to stay in control and make Russia stronger. The people backing him who keep him in power will only do so for as long as he looks good. If he starts a war with the West, even a win would be so costly as to see him deposed, and I don't think Russia could win. A very bloody stalemate, perhaps. Nuclear arsenal is essential to their prestige, but Russian thinking about use of nukes has always been different to the West.
As for Korea, I think the very worst case scenario would be a limited nuclear exchange. Messy, but very one-sided.