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Is it possible for a launch of a space object originating from Earth to go so badly that it would cause permanent damage to the Solar System or even destroy it?
Example: a spaceship going off trajectory, ramming into a small planet or satellite, and destroying it or permanently changing its orbit or gravity, therefore changing the flow of light and radiation throughout the entire Solar System.
First, some perspective.
The impact of a single fragment of the Shoemaker-Levy comet on Jupiter released an estimated energy equivalent of six million megatons of TNT (approximately 600 times the size of the world's nuclear arsenal), leaving an impact scar that was visible for several months afterward. That was obviously a bad day for Jupiter, but that impact had little to no effect on anything else, including Jupiter's moons.
We simply cannot build anything that could deliver the kind of energy necessary to cause any kind of damage to anything bigger than a small asteroid (where "small" means on the order of a dozen meters across). We smacked a hammer into a comet which excavated a lot of material, but didn't significantly alter the comet's path. We could possibly knock a larger asteroid on a collision course with another planet, but that wouldn't be an accident - it would require planning and a non-trivial amount of energy.
Objects in the solar system are so massive and so far apart that nothing we could launch would have much of an effect.
No.
Look at the numbers:
10000 kg. => Your rocket
73420000000000000000000 kg. => The moon
5972200000000000000000000 kg. => The earth
1989000000000000000000000000000 kg. => The sun
To put this to scale, the rocket has a similar mass towards the moon as that of one human cell towards a human.
You can throw it as fast as you want, it won't have any effect.
Not in terms of physical damage, no. Rockets and spaceships and all the energy they contain is just so very, very, very, very small compared to everything.
There is one possibility for large scale damage though, even if it is very unlikely. There is a very small chance that some kind of virus or bacteria carried on a vehicle to some destination might just happen to survive and spread when it gets there, and if so, that could infect, pollute or otherwise damage where it landed. If you were really unlucky, over a very long time such an infection could spread to cover the entire body it landed on.
Although it is very unlikely, NASA do take this quite seriously, to the point of sterilizing all spacecraft components and assembling them in an ultra clean room, as well as calculating the probability that organic life survives that process, and survives the flight time to the destination, to make sure that the chance of contaminating the destination is sufficiently low. The Office of Planetary Protection is responsible for this.
Let's assume you have a rocket made of anti-matter. Let's give it a 100 metric ton rocket of the stuff. $e=m c^2$, and with anti-matter, we are converting a mass twice the size of the rocket to pure energy. Let's just say that they have answered that question on World Building, you are quite a bit short of the limit of $10^{15}kg$ required to completely destroy the Earth alone, let alone the entire Solar System...
One idea that came to mind is near-earth asteroids. There have been discussions on how to nudge one away from earth, but if you wanted to switch to evil genius mode, you could nudge it towards earth - Wouldn't destroy the solar system, but would make for a bad day/year/millennium/... for earth.
For a launch, I'd say no. About the only scenario where it could be bad, at the planetary level, would be if the mission was to go out, say past pluto, and redirect a comet. And something goes wrong, and you end up aiming the comet at some planet.
Even that would take a lot. But if we posit that the original mission plan was to move the comet. Just that the execution screwed up, and moved it the wrong way -- then maybe.
That still wouldn't affect the whole solar system. It would ruin your day if it hit the planet you were on. But the rest of the planets would be fine.
unless you want to talk about timescales of billions of years. Then.... maybe ... just possibly... you could get a butterfly effect. where your ship slightly changes the orbit of hundreds or thousands of pluto-distance comets and over time, we get unlucky enough that several planets get impacted, maybe several times.
so, maybe earth (well - Life on Earth - the rock itself wouldn't notice), the moon, maybe some bases on a few other moons, would be toast?
as far as hurting a gas giant, or the sun. NO. Not Gonna Happen. There's an XKCD What if? comic about “what are the chances of accidentally affecting Jupiter's orbit?” Suffice to say, you cannot accidentally affect a gas giant in any meaningful way.
