Let me try to explain all this phenomenon.

About the wave thing. First of all the wave and an EM field are abstractions of certain effects. Now try to picture this. I have two charged particles.

The two dots

And they have fields of their own. When one of them oscillates up and down, it creates a so called distortion of its field and that distorted field affects the other particle. The oscillation energy of the first one is thus transmitted to the second one.

Physics is not interested in "WHY it affects the second one" it is interested in "HOW these two particles interact" the solution is an EM wave. You abstract (reduce) this effect to a wave (due to mathematical solutions).

At times when Quantum mechanics first emerged, it was obvious that physics actually observes the interaction of two physical systems. You cannot look at something and say that "it does this". You need to observe the effect via a subject and the deduced conclusion is the "how" of this effect.

The wave particle duality is a matter of observation.

Imagine that you are to observe me with closed eyes and to find where I am you throw me a giant rock. You will calculate my position from the "aaah" sound, but what you have calculated is my position before the rock hit me. After the event I am far off. Now photons, electrons are tiny objects. If you try to observe them you disturb them and you can only get a conclusion with an uncertainity.

The rate of this uncertainity determines if you will observe the subject as a particle or a wave. A high energy EM wave will cause a localized effect thus observable as a particle; whereas a low energy EM wave's effect will not be that localized, so you observe it as a wave.

It is nice to see a fellow physicist. But you are wrong on this. Speed of light is same everywhere. You observe it as if it is slowed because the effect is constantly absorbed by the particles in the dense matter. The light is constantly absorbed and re-emitted. This slight delay causes the change in the permeability and the permittivity (epsilon and mu)of the matter in question.

If one considers everything as observation one will not be puzzled about Quantum Mechanics and Relativity. Quantum Mechanics considers small objects as I have mentioned above which then comes up with the uncertainity principle and wave-particle duality.

In the case of special relativity. Stop thinking about Lorentz Transfomations and first consider this. A dead star far away is still observable from here as if it is there. It is because EM waves (light) has a limited speed. So for anything I observe, I cannot observe them instantly, there is a time delay for the effect (light) to reach me. And the faster my observable moves the bigger the effect. Lorentz Transformations correct this issue.

The problem with special relativity is that calculations are possible for only inertial referance frames (observables must move with constant speed and must never accelerate).

The problem with this picture is that, this picture assumes the return of one of the twins. Either it makes a circular trip or decelerates and accelerates in the opposite direction, it does accelerate. (for non physicists: Circular motion requires acceleration). Thus the relativistic calculation is not valid here. Thus not making this a paradox.

It is true that if one of the twins move with constant speed close to the speed of light away from the other twin. The moving twin will observe that the standing twin stays younger. But that is only what this twin OBSERVES. The image of the other twin takes time to reach the moving twin, just as the image of the moving twin takes time to reach the standing twin. hey both observe the other one younger, but neither is true. They both get old at the same rate, it is their observations that is deceiving.

These are actually very simple concepts when you know the underlying logic behind. I have seen many professors who will present them as magical, mysterious phenomenon; but that is not the case. As long as you know how to triangulate phenomenon, you see that it is all logical and even intuitive.

Just think about this:

When you do an experiment you are in the experiment too. The physical effects should reach you too, so you cannot exclude yourself or the detector you are using for the experiment. And inevitably the detector or you do get affected by the events to produce a result thus affecting the experiment itself.

After knowing this tell me which one is more intuitive. Quantum Mechanics which (simply put) accepts that when I throw a giant rock at somebody, I must affect the subject of the experiment. Or classical mechanics which considers that "when I do an experiment I observe the absolute event without even affecting the event a tiny bit"

So which one is more intuitive.