The pulse from the surface would be a bit slower than once every ms due to time dilation, and the pulse from the center of the earth would be even more time dilated than that.

Time dilation comes from a speed difference or the gravity gradient you travel/observe across. The pulses of light don't observe any gravity at the core itself, but once they start traveling out from the earth, they feel the gravity from whatever portion of the earth they've traveled out from, and would appear proportionally time dilated.

If the earth were a big hollow shell like an inflated ball, two observers anywhere inside the shell would feel no gravity due to the mass of the earth, and would not observe any time dilation when observing one another (even if one was at the geometric center and another was hanging onto the inside of the shell). But an observer outside the shell would feel the entire gravity of the earth and would see both people inside the shell as time dilated by the same amount.

No, because light and information have to cross out of the gravity well to get to an observer, and time dilation is always measured in reference to some other point, in this case presumably outside of the core of the star.

Also the shell theorem shows that there is no net gravitational force at the center of a sphere. Lots of pressure down there, but the mass on all sides of the core cancels the gravity out.