would it be safe to dispose of nuclear waste in the earth's molten core?

It would work, the problem is accessibility. Subduction zones tend to be deep oceanic trenches, so you would have to get the waste down there then drill a hole in the seabed to put it in.

You could probably just drop encapsulated waste into the ocean over a subduction zone, which would work as well, but the environmentalists would never let you get away with it. Also, shipping huge amounts of radioactive waste by sea is not a good idea, sooner or later there would be an accident.

The plates move so slowly in human terms that by the time it has been subducted we will either have ceased to exist as a species or we will have figured out a better way to deal with it, so we might as well keep it where we can get access to it if need be.

Not plausible, unfortunately. Tectonic plates slip on the order of only a few centimeters a year, and that is if they are really fast! This movement wouldn't allow for rapid burial of nuclear waste.

Also, the deeper you go into the Earth's crust, pressure and temperature increase, any capsule you might use to store this waste would most likely melt away, along with your concentrated radioactive waste.

When we do dispose of nuclear waste, we actually bury the waste far away from faults in order to avoid groundwater contamination.

If there is a weakness in the earth's crust (faults, fractures, or joints) kinetic energy is often transferred along these areas. This is also the case with hydrothermal fluids. In reality, it would be dangerous to place nuclear waste near or on a fault!

What if at some distant day interior the destiny we could have the technologies to go away earth and colonize the universe, yet by using the time we attain this point of progression we've created such an air of secrecy of rubbish around the earth that it may be risky to attempt to fly by it? save it on the earth. Are we actually going to be a planet to is encapsuled in its own crap?

I doubt that's safe. It's essentially a nuclear meltdown. Sending the waste to the core would distribute the radioactivity everywhere.

You could put the waste in your garage. In a billion years a tectonic plate will slide it under another one.

Safe, perhaps, but what if we find an effective way of using/recycling that waste and need it someday? There is only so much radioactive material available on Earth...

it could be very dangerous because, while the nuclear waste increases the probability of it reacting also increases. which could cause major earthquakes, canyons, new volcanoes, and countless other disasters, such as methane pockets bursting open, and dormant volcanoes suddenly not dormant anymore.