(2) When you have a loose charge that is not oscillating, you have virtual photons being released which have and electric field vector pointing toward the charge (by convention, I use blue for negative and red for positive here)
(3) Regardless of the convention that is used, Opposite Virtual Photons traverse through each other and lauch the opposing charge in the opposing direction of travel. Like Virtual Photons bounce off each other and also off of like charges.
(4) If the Charge is accelerated up and down, then that causes the virtual photon to manifest as a real photon with an oscillating electromagnetic field in the plane perpendicular to the oscillation.
Update:
If you are looking for a more accurate mathematical explanation of photons as relating to Maxwell's Equations, relativity, and quantum mechanics then check out my other post Photons and Parametrics.
If you are looking for a mathematical explanation of photons as relating to the index of refraction/absorption then check out EM Frequency and the Index of Refraction
If you are looking for a mathematic explanation of relativity as it relates to time dilation (eg. redshifting frequency for wave particles) then check out special relativity
