Regarding Hawking Radiation, if a particle - antiparticle pair forms with the particle within the event horizon, wouldn't the black hole increase in mass and appear to emit an antiparticle? Are the chances of the particle forming within the event horizon equal to the chances of the antiparticle forming within the event horizon? If so, wouldn't the black hole have a net change in mass of 0?
Pair production by gravitational field of a black hole must respect energy conservation law. So the total energy of the pair must be zero. But if a particle escapes away from the black hole its energy must be positive (there are no negative energy states in the asymptotic region). For nonrotating black holes all states with negative energy are located inside the event horizon. So it would always be negative energy particle that is absorbed while positive energy particle that is escaping away. As a result black hole mass would be decreasing.
For a rotating black hole there are negative energy states outside the event horizon, in the ergosphere region of a black hole. Thus it is possible for positive energy member of the pair to be created inside the horizon while the second member of the pair would occupy negative energy state outside the horizon. This is the closest to what OP termed “inverted Hawking radiation”. And since the negative energy particle is not escaping but is staying near the black hole, the total mass of the system as felt by an observer far away would remain the same, only now the black hole proper would be heavier but would be surrounded by orbiting matter with negative energy.
What occurs further depends on whether the particles are bosons or fermions. For bosons it is possible that particles already orbiting the black hole would stimulate production of more and more such particles (just like stimulated emission of coherent photons). As a result, we would have a black hole surrounded by exponentially increasing perturbations of the boson field, so-called “black hole bomb”, that would eventually destabilize rotating black hole (see here for a popular account of the effect). If the particles forming are fermions, Pauli exclusion principle would prevent instability, and the black hole would eventually form what is called “Kerr–Fermi sea” (a popular account could be found here) when there is a filled shell of fermions extending well past ergoshere.
Since only massive particles can have negative energy modes around rotating black holes, and since Compton wavelength of all known massive elementary particles are much, much smaller than horizon radii of astrophysical black holes the effect of such “inverted Hawking radiation” for real black holes is tiny, even over long timescales over which ordinary Hawking radiation becomes significant, unless there exist ultralight particles weakly interacting with ordinary matter. In the latter case observations of black holes (such as gravitational waves from black hole merger) could provide evidence of such particles orbiting black holes (see another popular story).