How does a mirror "really" work? How would you use quantum electrodynamics to actually explain how a reflection occurs on a microscopic scale?

That's a interesting question but can be answered only using optics. Photons can interact between them making interferences (destructive or constructive). When they do, appears a interference pattern. But you don't notice those inferences. Why? First, to make an interference between 2 photons they must have the same wavelength. The color of the sources of light you use are white, then photons on all wavelength reach the mirror and you will have an interference pattern on red light, another pattern on blue light, another on green light, the same for the yellow, ... All interference pattern happens at the same time on all colors, then you don't see any interference pattern because you're seeing all at the same time (when you see all colors on the same time, you see the white color). But even with a monochromatic source of light you can't see any interference. Why? Because another condition to make a interference is the coherence. A real bulb is working launching photons randomly on all wavelength. Even on a monochromatic bulb photons with the same wavelength are launching randomly on all directions. To make an interference both photons must have the same direction on the same time,  and it's very difficult they do that because they have a random direction. That can be avoided moving away the bulb. Then all the photons you receive fom the bulb are on the same direction. But photons are emitting on pulses. Imagine a photon launching during a few nanoseconds from the bulb, the next photon is launching on the same direction few nanoseconds after but both photons aren't launching at the same time. They travel on parallel directions but one photon is on a forward position that another one and the don't coincide. Only few photons emitting at the same time and on parallel directions is making an interference but even when that case happen to make an interference a peak of one photon must coincide with a valley on the another one (or a peak must coincide with another peak to make a constructive interference). Then consider all the possibilities: two photons must be launched from a monochromatic bulb, on the same direction, at the same time and coinciding peak with valley. The probability that happen is low and the interference pattern is very, very, very weak. Then you see a normal image and superposed a very, very, very weak interference pattern. Result: you don't see any pattern, you only see the image. The second question: Why a photon don't have mass... Well, you must ask this to Good... Physicist are only "public notary" of the nature. They describe how the nature is but sometimes can't explain why is on that way but not on other. Today we know a photon don't have any mass because don't interact with Higgs field. Why? Don't know only we know that is on that way.

If you're doing QED you need to think in terms of Feynman diagrams. These are diagrams showing interactions between particles. Here's an example: Two electrons come in from the left. A photon travels between them and two electrons leave on the right. The electrons emerging have different momenta from the ingoing electrons. This is because the photon has carried some momentum from one electron to the other. When two particles interact in a way that changes their momentum, that's basically just another way of saying a force has acted on them. The rules of QED dictate what Feynman diagrams can occur in nature. If you're just considering electromagnetism then you can build any graph you like with any particles travelling along the edges. But the only vertices you're allowed must be three way vertices where a photon meets two identical charged particles. (You're also allowed the meeting of a photon, a particle, and its antiparticle, but in a sense this is actually the same diagram.) In a sense it's the defining property of electromagnetism. The crucial point is that there is no diagram which has two photons going in and two photons going out with different momenta. There is no vertex where four photons meet. (Or one where three meet, or five meet, and so on.) As a result photons don't interact with each other. They only interact with charged particles. Having said that, following the rules I described above you can construct this diagram: Note how it has two particles and a photon at each vertex. This diagram describes a way that two photons can interact by exchanging electrons in a complicated square dance. But it's an extremely unlikely interaction. It requires four events to happen rather than just one. So its strength is dwarfed by the strength of ordinary interactions of charged particles with each other. Nonetheless, there are some extreme situations where this kind of https://en.wikipedia.org/wiki/Two-photon_physics might be observed. For example it arises in astronomy and can be probed in particle accelerators. Using QED to explain why a reflection doesn't interact with the original light is probably overkill. But it's good to know that there is an explanation.

This is really too difficult to answer in a simple post, since you have insisted in the quantum explanation. The best explanation I have seen is by Feynman in his book QED, and Feynman takes almost  chapter. Given that Feynman is probably the best I have seen at explaining physics, and the book is really quite cheap, I suggest you try that out.

While absorbtion/emission of photons by electrons accounts for part of what can happen to photons encountering a material, I think reflection and refraction are not determined by electrons but primarily by the density of the material and the incident angle.Possibly, if the wavelength has to shorten over a certain percentage, the photon would reflect rather than penetrate the material.See microscopic explanation.https://en.m.wikipedia.org/wiki/Refractive_index# Also see the effect temperature and pressure has on the index of refractionhttp://nvlpubs.nist.gov/nistpubs/jres/67A/jresv67An2p163_A1b.pdf