What does this mean for the matter of Earth's core that is within this radius?
Nothing, since the mass outside the radius does not contribute to the force (see Newton's shell theorem).
A related question comes up for what happens when an almost black hole accretes matter and slowly becomes a black hole.
In the system of the coordinate bookkeeper the velocity of the infalling matter not only slows down but converges to zero when it approaches the Schwarzschild radius because of the gravitational time dilation. So there is never enough mass inside the Schwarzschild radius to form a true horizon in a finite coordinate time. The combined mass of the initial body and the infalling material will therefore be larger than before, but so is the volume over which the total mass is spread out. Therefore in the system of the coordinate bookkeeper the radius that contains the mass will always be larger than the Schwarzschild radius of the mass.
Prior to the moment of the Schwarzschild radius crossing the boundary of the object, what does the matter within the radius experience?
The observer that crossed the horizon in a finite proper time took an infinite amount of coordinate time to even reach the horizon, so there is no longer a connection to the outside world. He will also experience spaghettification before he inevitably ends in the singularity.