If a rogue asteroid close to the size of Pluto collides with Pluto, could this possibly cause enough disturbance to cause objects from Kuiper's Belt to be pulled into the inner orbit of the Solar System but without destabilizing any other Planets?
In the Kuiper Belt there is a whole lot of nothing.
A collision between a couple larger objects could easily distribute a bunch of their debris through the belt, and since Pluto and its moons are relatively large objects they could eventually bother something out there. But it is so large and sparse it could be a long long time coming.
It could also send mass sunward, Pluto is the size of the larger moons of the official planets. Neptune's moon Triton in fact seems like a reasonable candidate for something like this happening in the past; it is about the same size as Pluto and has a very odd orbit implying it was captured rather than formed.
If it happened to get close to any planet it might disrupt its moons, but would not likely destabilize the planet. But the solar system is pretty empty too.
Most importantly though it would melt as it came into the hotter parts of the system. Pluto has a lot of nitrogen ice that would evaporate before getting close to us. If it broke into little pieces before impacting it might even hit Earth without killing everyone.
TL;DR Pluto dying could be totally ignorable from Earth or other planets and not likely to be much of an issue to other Kuiper belt objects.
While many asteroids are composed primarily of rock and metal, most Kuiper belt objects are composed largely of frozen volatiles (termed "ices"), such as methane, ammonia and water. (Source)
Rather than creating a chain reaction of elastic collisions (like the opening shot in a game of billiards), it would create a short period of very inelastic collisions (think "snowball fight"). The result would be a lot of loose ice floating around the kuiper belt, but pretty much nothing else.
This started as a comment, but I think it deserves to be turned into an answer: Yes, a collision like that would significantly increase the debris coming in to the inner Solar System, and would significantly affect Earth, but would not destroy it or alter its orbit.
A collision of Pluto with a Pluto-mass object will be very energetic unless it's carefully set up not to be. (If it's set up to be a gentle merger, then nothing much will happen other than Pluto melts into an incandescent mess. I assume that's not the case you're asking about.) Anything like a head-on collision would be big enough to disrupt Pluto and scatter much of its mass as gas, rubble and fragments (mostly small fragments) in every direction.
Here's the catch: Pluto is in a resonance with Neptune which has the effect of stabilizing its orbit. If that resonance was disrupted (and most of the fragments would not be left in a resonant orbit) much of the bits would be scattered by Neptune on a timescale of a few to a few tens of Neptune orbital periods -- call it on the order of a thousand years
This scattering by Neptune would then result in the debris being scattered more energetically in all directions. Much would go outwards, but a non-trivial chunk of Pluto's mass would eventually make it into the inner Solar System of basically cometary orbits. (Some of it in multiple stages in the planetary billards game Jupiter likes to play.)
So you'd see an initial burst of debris from the collision followed by a continuing rain of bits and pieces for many thousands of years. It would be dramatically larger than what we see now, but it also would be dramatically less than the Late Heavy Bombardment. It would probably be survivable, but I suspect that Tunguska and Chelyabinsk-style events would become fairly common. (And no one would pay attention to the Perseids anymore.)
So the Earth would get a modest battering, but it's orbit would not be in any way disrupted. (Nor would the orbits of any other planet, nor would the orbits of smaller bodies in the Asteroid Belt or the Kuiper Belt. The collision debris simply isn't going to be big enough have significant gravitational effect.)
The impact as you described it would at worst tear Pluto apart and turn most of its ice in a cloud of gas that will disperse in space.
For Pluto to be dragged away from its orbit, you need the passage of a substantially larger object with a hefty gravitational pull. The timing should be just perfect for Pluto to perform a slingshot and then fall toward the inner solar system.
No. First off, the impact would evaporate the frozen nitrogen and water ice forming a good part of pluto ( more on Pluto internal structure ) and a the rest of the Pluto+asteroid mass would be scattered into relativly little chunks.
Those pieces probably won't impact any Kuiper body since the Kuiper belt is very big and mostly (as in >99.99%) void space.
They also won't have enough gravity to provoke disturbances to the belt since the only disturbance we know of has been triggered by Neptune (which is far, far bigger) and it still didn't pull any object near the inner system, it just made their orbit very outwardly eccentric).
Studies since the mid-1990s have shown that the belt is dynamically stable and that comets' true place of origin is the scattered disc, a dynamically active zone created by the outward motion of Neptune 4.5 billion years ago scattered disc objects such as Eris have extremely eccentric orbits that take them as far as 100 AU from the Sun
Source on the nepune event: https://en.wikipedia.org/wiki/Kuiper_belt
The Kuiper belt is a huge huge huge region of space with a few small and scattered dwarf planets and asteroids. There is not much gravitational interaction between the different dwarf planets.
However, Pluto has 5 moons. If Pluto collides with an object of similar size, then that collision might result in Pluto losing some of those moons. They might get launched towards the inner solar system.
The largest one of them (Charon) has a mass which is about 1/2500th of Earth. If it would collide with Earth (a very unlikely coincidence, but it might make for a good story), then that event would not alter the orbit of Earth much. But it would extinguish all life on it.