(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)
(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.
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