Is a planetary collision cascade possible? It would seem to go against common sense that it would be possible for a small rocket to destroy a solar system, but do the laws of physics actually forbid it?  Suppose a rocket were to hit an asteroid and knock it off course.  Then this collides let's say with a small moon and it falls into the planet changing it orbit eventually leading to a cascade of unimaginable destruction?
 A: The scale of energy of the rocket is many, many orders of magnitude less than that of the gravitational energy of planets in a solar system.
The only way that a small energy input could make a significant difference is if the system were already on the edge of instability.   So it's possible in the same way that a mouse could knock over a house if you had a Rube Goldberg mechanism that could be triggered by knocking over an initial domino.
Any such system would be unstable on long time scales even without the rocket.

Can it be proven mathematically impossible? Maybe a stable system could be destabilized but it would take longer? Even the order of eons, but still happen?

First of all, outside of a binary system, it's difficult to prove any sort of stability.  Our own solar system is mostly stable on timescales of a billion years, but simulations beyond that can show major disruptions.
Because of this, the long-term trajectory of a multi-body system is not possible to know in detail far into the future.  As such, it is also difficult to say what effect a small disturbance (such as a rocket), ultimately has.  We can't distinguish the outcome where it is present with the outcome where it is missing.
If an enormous mass like a rogue black hole wanders through the inner solar system, everything gets scattered and it's trivial to tell the effect.
If a rocket moves through, then we can (with advance notice) possibly say that some body in the system had the trajectory changed by some tiny amount.  But for such small energy changes, we cannot say that it had any appreciable affect.   It cannot take a nearly stable system to an unstable system in a short time.  If a nearly stable system becomes unstable after a long time, there's no way to pin the rocket as the "cause".  Maybe it would have happened anyway.  Maybe it would have happened earlier without the rocket.
A: A single planet orbiting a star is stable: it follows an elliptical orbit, and if you nudge it slightly you get another elliptical orbit.
Two planets, or a planet and a moon, is a different matter. This is the celebrated 3-body problem, and for some combinations of masses, positions and velocities it is chaotic in the mathematical sense: a tiny nudge of starting state will over time grow exponentially and make the system end up in a very different state from where it would have been. The issues are (1) not all configurations are chaotic, many just follow quasiperiodic orbits, and (2) the chaos is typically bounded, so that the planets do not careen off into space (see the KAM theorem). Still, in this case a tiny nudge can in principle cause a big change.
The full n-body problem is like this but even more complex. Proving that the solar system is stable was a major challenge and the end result is largely statistical. Still, most bodies are in well-behaved orbits and are not too chaotic. To cause a cascade you hence need to cause a fairly big disturbance of everything rather than a nudge.
Technically, I think the nudge need to be able to move the system out of one of the invariant torii into a chaotic region connected to infinity. I don't think I have ever seen a Poincare plot of the full solar system model, but I doubt there is such a chaotic region nearby.
