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These questions need answers

If you, hypothetically, could make a hollow spot in the core and keep it hollow (and not burn up, and somehow breathe, etc.) then you would feel weightless inside it, yes. There would be some tidal force heating effects but nothing you'd notice unless you started moving really fast.

Somewhere in the upper mantle, though, you'd experience slightly higher than surface gravity while also being pulled in all four directions at once.