Yes, they're actually planning on doing this.
They also plan to do the same thing to Earth. "The Core" was just a ploy to lull the public into a false sense of security. They are going to detonate a quantum hypernuclear bombonoid in the center, and kill everyone.
Anyway.

No, if Cassini goes into Saturn's atmosphere, nothing will happen. If memory serves, Saturn has a lot of helium in it, and it's just not massive enough to start a fusion reaction.
Galileo plunged into Jupiter's atmosphere, and it's mostly hydrogen (again, if memory serves me correctly), plus it's much more massive than Saturn. Again, it's like spitting into the ocean and expecting something to happen.
People who know NOOOOTTTTHING!!! about science get all, "OMG Plutoniums!!!! Saturn has hydrogen and it's big! Nuclear things happen! It'll become a star!"
No. Stars don't get "ignited" by some radioactive element or compound starting them up. Mass builds up, increasing pressure in the core. This increases temperatures, which makes nuclear fusion easier to achieve. In order for a star to be stable though, it needs to have sufficient mass to keep the immense energy generated by fusion to keep from blowing the new "star" apart in a giant puff of plasma. It's an equilibrium between gravity and the star's own energy.
Saturn isn't massive enough to do this - it can't generate sufficient heat or pressure inside its core to start a fusion reaction. A few kilograms of plutonium isn't going to do a damn thing to start a fusion reaction, either. You could even detonate a fusion bomb in Saturn's core, and you might not even see the ripple by the time it got to the surface. The fusion bomb isn't going to start a full-scale nuclear reaction in Saturn's own atmosphere or core.
Nothing to worry about. We can't spawn new stars, at least not until we have the technology to funnel trillions upon trillions of cubic kilometers of hydrogen into one spot in space, and do so rapidly enough such that its own gravity will hold it together before it dissipates.
You mentioned Google results, so I went looking. I confirmed my suspicions, that the people coming up with these theories of idiocy are in fact, you guessed it: idiots.
From here:
"1997- Cassini is launched for Saturn with a tremendous load of Pu-238 dioxide (72 lbs!), many times the amount actually needed to run the craft?s instruments."
No freaking kidding. When it's launched, yeah, it's going to have a high output. But guess what, that plutonium decays. By the time it gets to Saturn, the output is already going to be lower than at launch.
Let's say that it needs 100W to run, and at launch, the RTG puts out 100W. By the time it gets to Saturn, 7 years later, its output will decline. If I understand the equation properly, after that time, and given plutonium's decay rate, the RTG would put out only 94.6W. Oops, can't run the spaceship anymore. Damn, that was an expensive mistake. That, and they'd like to have the chance of an extended mission available, for something like what we have with the Voyager probes. Now, Cassini is in orbit around a planet, with various interesting moons - staying in a scientifically useful orbit consumes fuel for the thrusters, so this will likely be the limiting factor on its usable lifespan. It also lacks a scan platform, so the entire spacecraft must swivel in order to target and track objects for scans. This eliminated the expense and complexity of a scan platform on what was already the largest and most complex probe ever sent to another planet, but it came at the cost of increased thruster usage.
Given that there's been water (frozen, with perhaps an sub-surface liquid reservoir) on the moon Enceladus, they won't want to risk crashing Cassini into it, contaminating it. This is why they sent Galileo into Jupiter's atmosphere. It was not only running low on fuel, but its electronics were degrading due to the intense radiation at Jupiter. They didn't want to risk having it hit, and contaminate, Europa, a moon covered in a crust of water ice, possibly with an immense liquid water ocean beneath.
Concerning this "contamination" - spacecraft are sterilized, but there are limits to this, and specifications as to just how sterile something can be. Really making sure that
every last microbe was dead would also likely nuke the electronics, and would make it really damned expensive. So there might be a few unwanted stowaways on the probes we send out.
But anyway, yes, they send these things out with more plutonium than is needed. New Horizons - another example. It's got a 9 year trip to get out to Pluto. It'd kind of suck to have it get there and not have enough power to get the job done. Then it's going to continue to fly through the Kuiper Belt, possibly making close flybys of other interesting objects. Again, if it doesn't have enough plutonium on board, it won't get a hell of a lot done.
I'm sure that if you're setting out on a 100 mile drive, and your car gets 30mpg, you're not going to put exactly 3 gallons and 5.333 ounces of gas in the tank.
Summary: People making up this BS about turning Saturn into a star weren't paying enough attention during middle school science, high school science, or any science class.