Would a tachyon be able to escape a black hole? Or at least escape from a portion of the hole inside the photon horizon?
 A: Your question only really makes sense for a localized tachyon, i.e. one whose wavefunction in position space is constrained to a finite region of space (i.e. has compact support) because that is the only kind that will "fit" inside a horizon of a black hole. And the answer to your question for this kind of tachyon is that it cannot escape a black hole.
The edges of a localized tachyon fulfilling the Klein Gordon equation with an imaginary mass parameter can only spread at a speed less than $c$, even though the plane waves (momentum eigenstates) that make it up have phase velocities greater than $c$. That is, if we plot the region where the wavefunction is nonzero, the boundary of this region can only grow at a speed less than $c$. This is a consequence of the Paley-Wiener theorem, which shows that the Fourier transform of any function with compact support cannot itself have a compact support and must include arbitrarily high frequency components. This is elegantly summarized in John Baez's Article "Do Tachyons Exist?". The theory making use of the Paley-Wiener theorem is summarized in QMechanic's answer to "Do Tachyons Move Faster than Light?.
Since the disturbance of a localized tachyon cannot spread faster than $c$, it therefore cannot escape the inside of a black hole's event horizon. In concluding this, we also need to assume (or at least I will, because I only know classical General Relativity) that the inside of a black hole is exactly as e.g. the Schwarzschild or Kerr solutions (in the Kerr case we need to limit the angular momentum so that there is an event horizon) to the Einstein field equations would describe a black hole: no talk of firewalls or other speculative recent quantum phenomena. We also need to assume it is valid to simply transcribe a solution to the Klein-Gordon (or other suitable wave equation) onto the spacetime inside a black hole. So these ideas will apply for a big black hole, where the spacetime inside the horizon is of very low curvature compared to the scale over which the tachyonic field is nonzero.
A: Yes, it would. Tachyons are a hypothetical object that can travel faster than light. They also require infinite energy to slow down as they grow faster the more energy they lose. A tachyon positioned right may as well get stuck in the center for a Planck or two I don’t know but that would speed it up A L O T. Tltr: the closer a tachyon is to the center of a black hole, the faster it is.
A: Yes. Since faster-than-light travel is required to leave the black hole, and tachyons apparently propagate faster than light, such a thing would be possible. If tachyons existed... 
