Not necessarily. That's the sort of question the paper investigates, and it finds models for which fusion can continue in the star's core for an extended period of time.
> There’s no pressurized core at the center.
Yes, there is, because, as I noted, the matter falling into the hole radiates strongly, and the radiation has pressure.
No, it's not inches, it's about 3 kilometers.
But the holes at the center of stars that the paper is talking about have tiny masses, much, much smaller than those of the stars they are inside. Their schwarzschild radius could indeed be of the order of Angstroms.
The numerical simulations in the paper go on for a time on the order of the Sun's hydrodynamic time scale, which is tens of millions of years. After that time has elapsed, yes, the star could be completely consumed by the hole.