Frequency of light versus frequency of electron vibration I've been trying to understand photons and light, although entirely conceptually (layman with not much of a background here, but I really want to understand this a bit better) and there's a couple of things confusing me.

As far as I understand, when an electron "vibrates", it causes a change in the magnitude of the electric field that electron sends off into space.
This change in the electric field is felt by other electrons on other atoms sometime later (because light doesn't move at an infinite speed, it moves at some specific speed, which is the speed at which "changes in the strengths of fields at a distance are felt" move in, in our universe), and that causes electrons on other atoms to vibrate.
I'm not sure WHY electron $A$ vibrating relative to electron $B$ causes electron $B$ to vibrate...I was told by someone it had something to do with special relativity and the space that electron $A$ occupies becoming smaller...but it went way over my head. Anyways, I'm willing to accept that electron $A$ moving causes electron $B$ to feel a force and vibrate as well. But, when electron $B$ vibrates, electron $A$ stops vibrating...otherwise, energy wouldn't be conserved. Although what actually physically stops electron $A$ from vibrating...I have no idea...
We call the transfer of the vibration from one electron on one atom to another on another atom "light".
The higher the frequency of the originally vibrating electron, the higher the frequency of the electron that feels the changes in the field. This corresponds to higher frequency electromagnetic waves.

The above was just to explain my (low) level of knowledge on this. My questions are these:

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*When people talk about the "frequency" of light...is that the same thing as the "frequency" of the vibrating electron producing that light?


*Does one cycle of the vibration of the electron mean the production of one "photon"?


*If the answer to the above question is no (which as of editing this post, I think it is), then this is a follow-up question: Once an electron gets "hit by a photon" (whatever that means) and it starts "vibrating" with some frequency, for how long does it do that? I know it seems like an odd question to ask, but here's the thing - if the electron were to speed up into its vibration, and slow down, so that its frequency of vibrating changes throughout, then according to the equations I've seen that would mean the electron would emit light with multiple energies. But...that goes against the whole concept of quantization and photons, right? In short, if light is quantized, what does that mean in terms of the electron that started vibrating? Does it mean that the electron SUDDENLY starts vibrating and SUDDENLY stops, and when it stops it emits the last photon? Does the vibrating electron only emit a single photon?


*This is a follow up to the previous question - What is the connection between the "vibrating electron causes changes in the field it produces that causes other electrons elsewhere to vibrate" theory and the "excited electron sends off a single photon that hits another electron and excites it" theory? How do the vibrations and changes in field magnitudes correspond to the photons?
Or am I entirely and utterly confused?
 A: Electrons and photons are quantum mechanical entities, obeying quantum mechanical equations.
What you are describing with words is a classical description of light which is controlled by classical electrodynamic equations, Maxwell equations.

An accelerating charged particle produces an electromagnetic (EM) wave.  Electromagnetic waves are electric and magnetic fields traveling through empty space with the speed of light c.  A charged particle oscillating about an equilibrium position is an accelerating charged particle.  If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f.

Going to the quantum mechanical frame of photons , the classical light of frequency f emerges from zillions of photons of energy=h*f in a mathematically complicated way, The frequency is associated with the energy. An accelerated electron will radiate a photon according to the rules of quantum electrodynamics, and there will be a probability of generating this photon, mathematically complicated. This can be proven mathematically, but not in a handwaving way.
A: I'll have a go at a hand-wavy explanation which assumes that you are comfortable with the idea of an electromagnetic field. If you're not, let me know and I'll address that in an edit.
Light is ripples of electromagnetism. When people talk about the speed of light they mean how quickly the ripples travel through space. The frequency of light is the rate at which the ripples vibrate.
Most sources of light produce ripples with a mix of frequencies.
You can (very very loosely) think of the intensity of the light being the height of the ripple, so the more intense the light the more pronounced the ripple.
Experiments have shown that it takes a fixed minimum amount of energy to set off a ripple of a given frequency. That minimum energy is given by hf, where h is a tiny number known as Planck's constant (named after Max Planck) and f is the frequency of the ripple.
You can, very loosely again, think of a photon as the tiniest ripple you can make of a given frequency. If you want to increase the intensity of the ripple you have to build it up in units of a photon. So you can think of a beam of light as being the cumulative effect of billions of tiny ripples adding together to make a bigger effect.
Photons are (again loosely) given off by charged particles, in circumstances in which the particle loses energy which is transferred into the photon ripple. The frequency of the photon is given by e/h, where e is the energy taken from the charged particle (and h is Planck's constant again).
It's not right to think of the charged particle 'vibrating', so you can't think that the particle has a frequency of vibration that's linked to the frequency of the photon, although that is a tempting image and would be in keeping with classical ideas about electric fields.
Indeed, one the reasons that made physicists realise that there was something wrong with classical electromagnetism was that they imagined electrons orbiting in atoms like tiny planets, and classic electromagnetism said that the electrons would indeed create ripples as they orbited. That would mean that the electrons would be radiating off energy all the time, so they would soon slow down and spiral into the nucleus. The early ideas of quantum theory were that the electrons could only exist in certain orbits, in which they didn't create ripples, and that the ripples only happened when an electron 'jumped' from one orbit to a lower one. The energy given off by a single jump was the minimum needed to start a ripple, or in other words to create a photon. 
A: You don't need to go all the way into quantum mechanics to understand what's meant for "light frequency". 
Classically, light is energy being transported by the electromagnetic (EM) field. When the electron vibrates it modifies the EM field in such a way that energy propagates away. This field is a vector field, that means for every point in space the EM field has 6 values defining it, 3 for the electric field and 3 for the magnetic field. 
For simplicity, think only about the electric field for now. Every point in space is associated with 3 numbers wich determine the vector of the electric field there. As time goes by, those 3 numbers change because the field is dynamic so the vector that represents is constantly changing. In the case of monochromatic light this change is periodic. As time passes, the vector at any point may describe a circle, an ellipse or just get shorter and then longer like a spring. This periodic movement has some frecuency and that's what is meant by the light's frequency. 
For example, red light has a frecuancy of about $4.3*10^{14}$ Hz. This means that if you could measure the electric vector in some point in space, you'd see it oscillating $4.3*10^{14}$ times every second.
A: You are looking for a conceptual model that makes sense, so you want a classical explanation. Quantum explanations will not make sense that way.
You have the fundamental ideas straight. Opposite charges attract, same charges repel. It's a force field. A moving charge has a different force from a charge with no relative motion -- the direction and intensity of the force can both change.
It all goes in both directions, but let's look at just one direction. When the source charge oscillates, the force on the target gets bigger and smaller. A change in direction and intensity of the force. The change in force that's inline with the direction of the target tends to just average out, it gets weaker with the square of distance. The change of direction of the force stays the same, and the sideways part of it has one less degree of freedom, so it falls off slower with distance.
So over big distances, what you get is a little force that oscillates, moving the target sideways. If the oscillation continues for a long time, a target which is predisposed to oscillate at the same rate can absorb increasing amounts of momentum from the tiny force until it somehow changes state in an observable way. 
Alternately, if it is predisposed to vibrate that way, a little bit of force from this particular source might be just enough to push it over the edge to change state. The more force, the more targets that have almost enough will get enough.

But, when electron B vibrates, electron A stops vibrating...otherwise, energy wouldn't be conserved.

No, it doesn't work that way. But I'm not clear on how it does work. The source charge is sending out its force field everywhere in all directions, and some of it interacts with target charges while other parts of it doesn't. It has no way to know what will happen to all that force. Both charges could be stationary at time $t_0$, and then one of them gets moved around and then is stationary when the force arrives, and so they are affected by the force that was created at $t_0$ at different times and in different directions. It's a mess. Maybe we could get mathematical proof that the whole thing has to balance out. Maybe it balances out statistically, on average. Maybe part of the energy stays in the force field, and we constantly get radiation heading off into nowhere, with less and less available for anything else. I dunno.
Conservation of energy is true by definition. Any time we find conservation of energy being violated, we invent a place for the energy to go. For example, neutrinos. Energy was observed to disappear. It must have been carried off by an undetectable particle. Energy is observed to appear. An undetectable particle must have delivered it. We detect neutrinos all the time -- whenever conservation of energy is detected to fail in the right amount, it must have been a neutrino. A failure of conservation of energy (and angular momentum etc) what it MEANS to detect a neutrino.
When you have a system where you're sure about the energy, you can use conservation of energy to sort things out. Somebody says he has an automobile that runs by burning water? Conservation of energy says it's unlikely. But when you're sorting out the fundamental laws of the universe, you can't say "Particle A must have less energy because Particle B has more". For all you know, undetectable particles C and D might have interfered.

When people talk about the "frequency" of light...is that the same thing as the "frequency" of the vibrating electron producing that light?

Yes. Incidentally, other charged particles can vibrate to do that too. Protons can. But with more mass it takes more energy to get them to vibrate as much.

Does one cycle of the vibration of the electron mean the production of one "photon"?

No. Classically, a photon has the amount of energy it takes to get an atom or molecule to detectably change state. 

Once an electron gets "hit by a photon" (whatever that means) and it starts "vibrating" with some frequency, for how long does it do that?

Photon theory is basicly different from classical electromagnetic theory.
When you start with Maxwell's Equations etc, you get mathematical equations that predict how electric and magnetic forces work including electromagnetic radiation. But they can't predict the vagaries of atoms. They only describe the forces, and not the particular ways that atoms interact with those forces.
Photon theory explains some of the interactions of light with atoms. Atoms tend to absorb and emit light at a few specific frequencies, different for each element. 
Light can knock electrons off of atoms. It takes a minimum frequency to do that, below that frequency no electrons are lost. Above the frequency, electrons are lost even when the light intensity is low. The higher the frequency of the light, the faster the ejected electrons move. 
Also, we assumed that low intensity radiation gets absorbed a little bit at a time, over many cycles. So if it takes a million electron volts to knock off an electron, that's a frequency about 2.5*10^-20. So if we make the intensity of the light very low, so that it would take more than 10^21 cycles to add up to a million eV, it follows that no atom would get enough energy for at least a second and there would be a delay before the electrons popped off. But in reality, they start right away at a low rate. 
You might think that the obvious conclusion is that some atoms are primed and almost ready to go if they get enough energy at a high enough frequency, while other atoms are not. The lower the intensity, the fewer atoms that are that ready.
But they chose another obvious interpretation. Light does not travel in waves, but in packets of energy which can be absorbed pretty-much immediately. One atom sends a packet, another atom absorbs that packet. And then they waved their hands to come up with reasons that the packets traveling through space behave exactly like waves.
It doesn't exactly make sense. But soon the whole thing was replaced with quantum mechanics which isn't supposed to make sense. So that got settled.
Here is an expert explanation about photons.
expert on photons
A: Light is the way atoms exchange energy with other atoms or smaller subatomic particles while bonding or changing the level of charge. These exchanges of energy can happen at all the frequencies in the electromagnetic spectrum from gamma which is so fast/small that it will travel among atomic particles with out hitting one for a very long time. But when you have a gazillion of them the chances increase. And the have so much energy because they are fast that the have a big effect on whatever they run into, like causing cancer. At the other end of the spectrum are radio and broadband. Some of these wavelengths could exceed a kilometer. This is low energy light and there isn't much that it will not run in to or react with; it hits everything in its path. But because it has lower energy, its mostly harmless and have small affects. That's why putting a metal rod on top of your house can easily pick up radio frequencies. The light energy after being released from an atomic reaction always travels at the same speed away from its source in all directions in units to energy called photons. The number of wavelengths when compared to other photons at the same distance at that speed determine what type of light it is.
I think your understanding of electrons vibrating next to one another regards electricity. Electricity is created by causing electrons in the outer shell of certain elements like metal to vibrate by connecting it something that has large magnetic charge. Some elements which rapidly decay by either oxidation or some other readily avail natural reaction to other elements, are constantly releasing an abundance of radiation which gives them a magnetic affect or charge. When near an element like metal whose outer shell is susceptible to magnetic charges, the outer shell of electron begins to vibrate. That vibration travels like dominoes causing every atom next to do the same, like metal wire. No photons are lost or gained since the atom doesn't loose or gain a level of charge. Because the electron only vibrate and stay on the outer level, its a way of cheating power off an atom by getting it excited, but not excited enough to gain or loose a charge level. This is possible because of the strong and weak forces of atoms and how they hold together or break apart.
