Can a photon be absorbed by a proton? When incident light passes through a hydrogen gas, for example, does it have 50% chance (since it's a 1:1 ratio of protons to electrons) of getting absorbed by the proton? Any chance at all? If no, why not? Does a proton have a "bandgap"? If yes, what happens when a photon is absorbed by a proton?
 A: A system can absorb a photon if the energy of the photon matches an excitation in the system. So the hydrogen atom can absorb a photon if its energy matches one of the frequencies in the hydrogen spectral series.
A proton is a composite object and it does have a spectral series. However the excited states of the proton involve rearrangements of the energy and angular momentum of the partons that make it up. The transition energies are in the 100MeV to 1GeV range so they are far beyond anything that could be excited by visible light. However gamma ray photons could excite such transitions and be absorbed.
A: 
Does a proton have a "bandgap"? If yes, what happens when a photon is absorbed by a proton?

For single protons, as in a plasma , there exists Compton scattering  .
The photon transfers part of its energy to the proton and scatters off at a lower energy/frequency, the proton taking up the energy-momentum balance. This is a continuous spectrum, from very low energies on.  
For high energy photons the quark structure of the proton can be  probed  and depending on the energy more particles  are created by the interaction with the constituent quarks. There are no energy levels for the quarks within the proton.
There do exist baryonic resonances, i.e. with the same three quarks at higher excited states. If one were to scatter gammas of the appropriate energy, some resonances will be excited , for example the N(1520) decays into a proton and a gamma so in the crossection a resonance will be seen passing that energy of center of mass in proton gamma scattering.
This is a theoretical study of gamma proton scattering at high energies.
A: The type of interaction depends on the energy of the photon, based on the Klein-Nishina formula.
From Wikipedia:


*

*Very low energy photons (visible light; as long as the photon energy
is much less than the mass energy of the particle, i.e. Compton 
wavelength) yields Thompson scattering, which is elastic scattering with electrons.

*Low energy photon (a few eV to a few keV, corresponding to visible
light through soft X-rays) can eject an electron from its host
atom entirely (a process known as the photoelectric effect), instead
of undergoing Compton scattering.

*High energy photons (comparable to the rest energy of the electron,
511 keV) bombardment results in the Compton scattering: the electron being given part of the energy (making it recoil), and a photon of the remaining energy being
emitted in a different direction from the original, so that the
overall momentum of the system is conserved. If the scattered photon
still has enough energy, the process may be repeated. In this
scenario, the electron is treated as free or loosely bound.

*Higher energy photons (1.022 MeV and above) may be able to bombard the
nucleus and cause an electron and a positron to be formed, a process called
pair production.

*Very high energy photons (hundreds of MeV) may yield Nuclear Compton
scattering.

A: The proton has to be absorbing and emitting photons for it be exerting an electrical force on the electron. Most of those are virtual photons, but clearly the proton is capable of absorbing a photon for some value of "photon" and "absorb".
