I have been reading similar discussions on how protons and electrons interact, using virtual photons and such. Yet, I can find little information regarding the subject in question. Can a beam of photons posses the characteristics of that of a virtual photon in a proton-electron interaction where the electron “thinks” there is a proton when there is not?
It is commonly said that the electric force is due to the exchange of virtual photons but you need to be very cautious about this statement. The electric force is calculated as if it was due to the exchange of virtual photons, but this is a computational device and virtual photons do not actually exist. There are no virtual photons being sent and received by your proton.
If you shine light at the proton there will indeed be an interaction. With high energy light we get Compton scattering or at lower energy Thomson scattering. But neither of these resemble the electrostatic force. The (real) photons do not produce anything like the electrostatic force because the electrostatic force is not transmitted by photons - it is just calculated that way.
No, you cannot. Virtual photons are mainly the famous Coulomb field, it does not propagate like photons to infinity.
An electron accelerates towards a proton with the exchange of series of virtual photons. In principle, you could produce the same acceleration by bouncing photons of the same energies off the electron with the same timings. It's the electromagnetic equivalent of the equivalence of gravitational and inertial interactions. Though at this scale the Uncertainty Principle makes it virtually impossible in practice.