This all assumes that humans are still vulnerable to these things when they happen. Assuming we still haven't developed the tech to literally control the environment, we may have advanced upon ourselves enough, either technologically or biologically to survive. If we enchance ourselves by either making us more disease resilient, or by replacing our bodies with machines, we can rule out disease, at least biological disease, someone could still make a computer-virus, but at the very least, we've eliminated one possibility.
it we make ourselves more resilient physically, we'll be able to adapt more readily to climate changes, cold or hot.
Of course, what the show doesn't take into account is that if we had a nuclear war, nature would get hit pretty hard too.
Still I don't think nuclear war would be the end of us all, this is a very big planet and nukes are really only going to fall on the major powers. The probability that someone will nuke Chile or the Congo is rather low. While we could still have nuclear winter, humans have survived the ice ages, so I don't think we'd all be dead.
Disease again is more likely to affect major population centers, and leave small tribal groups untouched.
I honestly don't think there's any disaster that would destroy every last bit of humanity. I think most "civilized" nations would be gone, but considering that we can grow plants in water under artificial light indoors and we're on our way to growing meat from stem-cells, I think a good portion of "developed" society could withstand most disasters.
"So if you go to Washington, it's buildings clean and nice. Bring a pack of matches...and we'll burn the White House twice!"
"Nobody's talking about extermination. No one ever does. They just do it." - Magneto
"Don't solicit for your sister, that's not nice, unless you get a good percentage of her price."