What *is* Magnetism? Short variant: What is Magnetism exactly?
Or, longer variant;
As an amateur science/(hard) sci-fi enthusiast, I'm slowly but surely realizing that Magnetism feels like an "odd-one out" in the gallery of building blocks of the universe, at least Gravity has a relationship to mass and to particles themselves, but Magnetism seems to behave in pretty arbitrary ways in relation to that.
To put it another way; Magnetism, or Electromagnetism, has a few properties which seem "odd" to me;

*

*Right-hand rule, that magnetic fields rotate clockwise around a positive charge (when viewed in the direction the charge is going)
Why clockwise? I thought most things in science would have a symmetry
or balance, this strikes as odd as such.


*That magnetism isn't further reducible, but seems to be a constant fundamental "force" or "property" of space, similar to Gravity and the weak/strong forces (at least to me).
Is there any explanation as to why Magnetism exists, alongside these other forces?


*Lastly, I see a lot more hubbub around Quantum Mechanics and how Gravity doesn't fit in there, but I see almost no explaination or hubbub how Magnetism (which I think could be classified as a "classical" force up to that point) fits in Quantum Mechanics.
Does Quantum Mechanics explain (on the smallest scale) how Magnetism emerges or "fits" into the whole?


*Magnetism works in the vacuum of space, and in other answers I saw light waves/photon particles linked into it. The part of photons/light being related to Magnetism interests me the most, I know light are radio waves (but displayed in a particular spectrum), but does that mean that Magnetism is the carrying "thing" for light and radio waves, essentially? What properties emerge from this that're fundamentally consequential to other parts of physics? (I.e. what is the relation between c and Magnetism here? is there any?)
In short; Magnetism seems pretty weird to me when compared and placed next to the rest of the gallery of physics, even though it interacts properly with it, to me it feels like the odd one out, am I alone in this?
 A: *

*Right-hand rule is only a convention. We also have the right-hand rule for a positive oriented coordinate system but there is nothing that says we couldn't have used our left hand. We "need" to use either right or left and it happens to be right-hand.


*Magnetism is not a fundamental force. There only exist 4 fundamental forces, electromagnetic force, weak force, strong force, and gravitational force. However one could talk about a magnetic force as explained by Lorentz force law but we will come more to that in (3). If you perhaps meant electromagnetic instead of magnetic then there is no good answer. Your question could also be "Why does the universe have matter?", "Why does Newton's second law hold?" or more specific to the electromagnetic area "Why does the Biot-Savart Law hold?" both Newton's law(s) and Biot-Savart law are taken as true, also known as postulates. We just happen to live in a universe where these are true and asking why this is the case is leading more towards philosophy than physics.


*If your question were relativity instead of QM then it would be quite interesting since the reason that magnetism occurs is due to charge in motion. And as soon as we have motion there could be something going on with relativity and that is the case here. If you were to have two charges in empty space. Imagine one being still and the other one moving, then the one standing still thinks that the one moving is the one who is creating a magnetic field. Now you might think, "but wait how do we know who is moving and who is not moving?". The charge that is moving could easily think that it is standing still and that the other one is moving. This is relativity. Back to QM; For the following have in mind that moving charge creates a magnetic field. From the simple atomic model, one can think of the electron (which has negative charge) as moving around the nucleus. Notice that I said "moving around" you might now ask "does this create an electric field?" and the answer is yes. Now QM is really useful to explain and understand since we are dealing with single particles on a tiny scale. Someone with more knowledge of this question may gladly elaborate on this.


*I'm a bit unsure about what the question is but I will try to be close to what it might be. Light is as you almost said electromagnetic waves. The electric field and magnetic field (which are perpendicular to each other) oscillate back and forth, see picture for clarification. So the relation between $c$ (which I assume you mean light) is that magnetism or more precisely magnetic fields along with electric fields is what light really is.

A: 
Notice: This answer may be inaccurate or open do debate. Please read the comments below

The oddness that you feel is a consequence of the fact that magnetic force may not really exist. It is a consequence of our natural tendency to believe that the universe obeys Galilean transform.
Consider a test charge moving with respect to us and a charge density at rest in our laboratory.  By observing the test particle trajectory, it is noticeable that it is different from what we expected to be, if only the electric force created by that charge density were consider. By Newton’s laws, we have to assume that another force depending on the test particle’s speed should be added to account for this excess push, a force we would call “magnetic force”.
Now consider the frame of the test particle. For an observer moving with it, there is no movement and therefore no magnetic force. However, the electric field measured by this observer will be different than the one measured by the first observer because this one observes a bigger charge density than the first one. The end result will be both agreeing on a stronger force that just the first observer’s electric field, even though both will credit it into different phenomena. The first one will credit the excess force into a “magnetic field” and the second one into a higher charge density due to length contraction.
Now the question is. Who is right? What is physical? A magnetic force or length contraction? It seems that assuming one of them as real and the other as a tool, lead us into the same empirical results in both situations, so, I leave to you the responsibility of finding out the solution to this problem.
For a visual description of what was told check this video, especially from minute 6:30 onwards.
