A black hole is believed to contain a singularity with all of the mass as a single point. So this is well past the point of baryonic matter and in a region where our physics models break down.
If you just take the total mass of a black hole and divide it by the volume of the Schwarzschild radius (aka event horizon) you get a density MUCH greater than a neutron star. This isn’t a useful measure of the black hole density though, since all of the mass is at a single point of presumably infinite density.
Yep. In fact there’s a process called inverse Compton scattering that essentially works this way. In ordinary Compton scattering, a photon scatters off a stationary electron and typically leaves with less energy (since the electron gets a kinetic kick). In inverse Compton scattering, a photon collides with a moving electron which can cause the photon to gain energy.
One application of this is to produce gamma-ray beams. You take a beam of light (often from a laser) and collide it head on with a beam of relativistic electrons traveling in the opposite direction. In the electron rest frame, the photon has gamma-ray energy, while in the lab frame it might only be visible light. The back-scattered photon can then be boosted to the gamma regime in the lab frame, and now you’ve got a gamma-ray beam.