The Earth will not be destroyed by an asteroid or comet impact. There just aren’t any asteroids or comets around any more that are big enough to do that. The really big rogue bodies, like the Mars-sized object that collided with the Earth to form the Moon, either collided with something or were ejected from the solar system early in the history of the solar system.
The objects we’re worried about for extinction events are probably in the 1-50 km diameter size range. There are a fair number of those still around. Some famous examples are Comets Halley (about 10 km in diameter) and Hale-Bopp (40-80 km in diameter). There are also Earth-crossing asteroids in that size range. We think you’d need an asteroid or comet at least 1 km in diameter to have global effects. Something smaller than that could have devastating local effects, but we don’t think it would affect the global climate too badly. Of course, we have no historical data on any such impacts, so this is not certain.
Whether we could do anything about an incoming asteroid or comet depends on how much warning we get. It’s possible that we could discover one that was headed straight for us a couple of years before it impacted, in which case there’s pretty much nothing anybody could do. We discovered Comet Hale-Bopp about two years before it got to perihelion (its closest approach to the Sun). Fortunately, it wasn’t on a collision course with Earth. You have more options of how to deal with a potential impactor the more time you have between when it is found to be on a collision course with Earth and when you think the impact will be. It’s generally preferable to deflect an incoming asteroid or comet so that it doesn’t hit Earth, rather than trying to blow it up. If you blow it up, and all or most of the pieces hit Earth, you are still getting the same kinetic energy from the impact, so you haven’t really improved the situation, despite what you see in some movies. Of course, it’s going to be easier to deflect a smaller object than a bigger one.
I’m an astronomer, so I don’t really know what the chances are that humans would survive something like the K-T impact. I’ll speculate that most humans would be killed. Your best bet for surviving is probably to be underground when the impact happens. It would probably cause an impact winter, which would mean growing any crops for food would be difficult for a while. The global firestorms that the impact would likely cause would wipe out all crops that were growing at the time of the impact, as well.
I’ve seen various estimates of how likely you are to die from an asteroid impact. They generally range from about one in a million to 1 in 75 million.
We don’t know what caused the Great Dying. An asteroid impact may or may not have been involved. Same goes for a lot of other mass extinctions. Part of the problem is that, if there are impact craters from asteroid impacts, they aren’t always in places where we can find them. They could be on the ocean floor, and be very difficult to find. The ocean floor gets recycled- some oceanic crust gets subducted into the Earth’s mantle, and new ocean floor gets created at mid-ocean ridges. If a crater was later subducted, there is of course no way we could ever find it. Even if it were still there, finding it might not be easy. The disappearance of Malaysia Airlines Flight 370 showed how difficult it is to search for something on the ocean floor when you have no idea where it is. The impact crater would be bigger than the wreckage, but it could also be anywhere in the oceans, so it wouldn’t necessarily be easy to find. It’s not even always easy to find really big impact craters on land. There is one that was hiding more or less in plain sight in Germany for many years.