You very nearly got there! Let’s run with your example of a small room at the center of the Earth, but for my sanity, I’m going to make your room a sphere instead of a square, because everything else involved in this example is going to be round, and it makes the example easier.
So; gravity pulls on any object with a force that’s related to the masses of the two objects involved, and inversely related to the distance between them. This tells us that the more massive the two objects, the greater the pull, and the greater the distance between them, the weaker the influence. For us, on the surface of the Earth, we can work out how strong gravity is. All you need is the mass of the Earth, the mass of a human, and the distance between the center of the Earth and the surface of the Earth.
The mass of the Earth in kilograms is 5.972 × 10^24. 10^24 is a septillion, which is a number so outrageously large that it might be more manageable to think about as a trillion trillion kilograms. (In SI units, which most physicists use, this is gives you the prefix yotta. 5972 yottagrams! It’s fun to say.) The mass of a human, in comparison, is negligible. The radius of the Earth is 3,959 miles - 6,371 km. If you plug these numbers in, you pull out the gravitational acceleration at the surface of the Earth; 9.81 meters per second every second. This pulls you in towards the surface of the planet.
Now, we’ve done something sneaky here, which is to assume that we can place the entire mass of the Earth at the very center of the Earth, and consider ourselves simultaneously at the surface, and six thousand kilometers away from the Earth’s mass. We can do this because the Earth is a sphere, and that means that there's an awful lot of symmetry to work with. You can also do the math very carefully, considering the pull of the Earth’s mass to the left of you, which will pull you slightly to the left, and the pull of the Earth’s mass to the right of you, which pulls you equally strongly to the right. There’s no net force going sideways, because you’re standing on a symmetric planet, and all the left and right directions will cancel out. All that’s left is the 'downward' direction.
Again, if you do it carefully, you have to consider the gravitational pull from the ground directly beneath your feet (which is quite close) and the ground on the opposite side of the planet, which is 7918 miles away. There’s the same amount of planet closer to you and farther from you (relative to the center of our planet), so on average, the force is the same as if it came from a point at the center. Mathematically, our trick of assuming that the entire mass of the planet is contained at the core of the Earth is identical to doing it all very carefully, and it is much easier to do.
What does this mean for your spherical room at the core of the planet? Well, this principle of canceling out forces if they’re pulling on you in different directions still holds, and so you’re absolutely on the money to say that you should be weightless in there. You absolutely could float at the center of the planet; the entire mass of one half of the planet pulling you to the left would cancel the remaining mass of the planet pulling you to the right. And the same is true of being pulled upwards/downwards, or any direction that you care to slice the planet in half. There would be no 'down'.
What does a depiction of space time look like in the middle of the planet? Let’s remember that our depictions of space time usually depict divots surrounding massive objects, where gravity pulls you “down” into the gravitational well. With no gravitational force, and no 'down', your room in the core of the Earth would have a space-time curve that was very flat. No bending, no vortex. If there’s no net gravitational force, there can be no slant or directionality to space-time.
It’s flat because you’re at the very very bottom of the gravitational well. To leave your room at the center of the Earth, you’d have to climb your way all the way back up to the surface of the Earth, and as soon as you left, there would be a net force, pulling you back down. As you climb out, more and more of the Earth is left below you, and the downward dragging force you would feel would increase almost continuously until you reached the surface. The surface is a much more hospitable place, in any case; it’s got all my favorite things on it.
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