Yes. You can think of it as: If you’re surrounded by an equal amount of mass in every direction (because you’re at the center of a set of spherical mass shells), then the gravitational force in every direction will cancel out.
Indeed, and as a related situation. You could be inside a black holes event horizon and experience zero gravity/acceleration. You just need two blackholes to cancel out the gravity.
Depends on the size of the black holes, spaghettification because of a high gravity gradients is a problem when you are close to the center of the black hole. Larger black holes make the much less of a problem, so you can cross the event horizon without any problems.
So if a cup of water were in the centre of the earth, and every layer of the earth above the centre wasn't there to compress it, then it would explode out!
If a cup of water at room temperature were on the surface of the earth and every layer of the earth above it wasn't there to compress it (so no atmospheric pressure) the water would violently boil immediately. If the cup is very tall or you're talking about a cup of very cold ice, the water's own gravitational pressure could keep the bottom stable, but to prevent the top from boiling/sublimating off, you'll need a lid to keep it pressurized.
Moving the experiment to a zero-gravity environment doesn't change the pressure considerations much.
If you somehow made a spherical hole in the middle of the earth and placed a cup of water inside the hollow (anywhere inside, doesn't have to be the centre) it will just sit there: Newtonian gravity is 0 inside a uniform spherical shell of mass. (Einsteinian gravity is probably mostly 0 but you'll get frame dragging effects if the shell is rotating I would imagine.)
You can do the maths to prove this yourself if you want.
Fill the hole with air at 1atm & it’ll sit there very slowly evaporating though, same as it does on the ISS or on the surface of the earth. Gravity is irrelevant.
You don't seem to understand where that 1atm air pressure on the surface of the earth is coming from. Hint: on the surface of the Moon there is none.
In the ISS the air pressure is artificially kept at a certain level. To make a proper comparison, you should check what water would do outside of the ISS.
The context of my original comment was someone expecting that a cup of water at the centre of the earth would explode due directly to gravitational effects. This is wrong, as I pointed out.
Even on the surface of the earth it is the air pressure that prevents the water boiling off instantly, not gravity itself. On the surface of the earth that air pressure is due to the effect of gravity on the atmosphere, but you can supply that air pressure by a number of other means in other places. The air pressure inside the ISS is not there due to gravity after all! I suggest to you that if we can posit a hole at the centre of the earth, then we can fill it with air at 1atm if we want to - we are in the realm of mathematical models here, not reality after all :)
>If you’re surrounded by an equal amount of mass in every direction
Which is a reasonable approximation for being 'out in space', but perhaps not an entirely intuitive one.