How can a photon (which has momentum) from one electrically charged particle to an oppositely charged particle cause these particles to be pulled toward each other - or how can a magnetic field cause an electrically charged moving particle to experience a force perpendicular to the source of the magnetic field if a particle with a non-zero mass moving between the two is the mediator of that force?
It is called Quantum Mechanics and to understand the processes a lot of graduate studies are necessary. A photon is a quantum mechanical entity/particle.
The classical magnetic field , electric field and electromagnetic radiation emerge from the underlying quantum mechanical level by the very large number of the elementary underlying processes involved.
The elementary processes are simple and can be calculated using feynman diagrams.
The calculation for electron proton interaction goes through simple Feynman diagrams which when calculated will give the behavior of an electron scattering off a proton, as a function of energy and angles.
The macroscopic interaction of charged particles emerges in a similar way that thermodynamics emerges from statistical mechanics. The underlying quantum mechanical processes add up to the observed classical behavior , but photons are not a useful concept macroscopically, in the same way that individual atoms in a gas are not a useful tool for the concept of temperature.