How does magnetism work? (Not asking what magnetism does) To clarify, if I asked "how does electricity work" I would want "electrons flow from a negatively charged area to a positively charged area", not "it powers electronics like phones" as an answer. The latter would be what I'd want if I asked "what does electricity do".
Onto the question!
Lets say we had 2 plates with 8 negatively charged balls each anchored in the shape of a regular octagon. They're basically completely identical. Plate A is directly above Plate B and both are still. Due to their negative charges, they repel. From a perspective where like charges repel this makes sense.
Now lets make both plates start rotating fast in the same direction. These plates are now creating a magnetic field that is pulling the plates towards each other. This force is so strong that the attraction caused by the magnetism equals the repulsion of the negative charges. From the static-electrical perspective of one of Plate A's balls, the plates would only act like this if net charge of Plate B's balls was neutral. From a perspective where like charges repel this seems strange and confusing.
Now lets make both plates start rotating really fast in the same direction. These plates are creating a strong magnetic field that is pulling the plates towards each other. This force is so strong that the attraction caused by the magnetism is greater than the repulsion of the negative charges. From the static-electrical perspective of one of Plate A's balls, the plates would only act like this if net charge of Plate B's balls was positive. From a perspective where like charges repel this makes no sense whatsoever.
Now due to the fact magnetic fields are created by moving charges, I'd say there's a good chance that the forces applied to charges due to magnetism are also caused by static electricity. Using this logic you can say the charge of Plate B would keeps getting more positively charged as the rotations speeds increased from Plate A's perspective. How would this work under the logic of "like charges repel and opposite charges attract"?
 A: There are many many misconceptions tied into knots in your question.
Firstly, the electric force between two charges doesn't depend on just the distance between them and their charges, it also depends on their velocity. When charge A moves then its electric field is different, so the electric force charge B feels is different. This means you can't reason from force to how much charge.
And actually charge is basically a count of how many protons and positrons and such you have minus out how many anti-protons and electrons and such you have, then multiply the result by a constant that makes an Ampere be a reasonable sized unit. It most definitely does not depend on frames or observers or perspectives. And it won't change unless you actually create or destroy actual particles.
Secondly when you have a rotating system of rotating charges, the charges only have instantaneously comoving inertial frames. So the frame of the charges you talk about is not inertial and Newton's laws don't hold in them.
Thirdly, you can't just assume objects stay rigid as you rotate them.
Fourthly, you can just assume that speeds less than lightspeed exist where magnetic and electric forces cancel. Sometimes one force really is stronger.
Fifthly, the rulers and clocks of the charges are also affected by the velocity. A great deal if they get close to lightspeed, so the above problem brings up this problem.
Sixthly, the Lorentz force a charge feels in its instantaneously comoving inertial frame is the electric force, so you don't need magnetic forces to describe the forces. But the electric field (unlike the charge) is not something all observers and frames agree on.
As for your title question, magnetic forces are more than just the Lorentz Force because their are intrinsic magnetic dipoles. But you have dipoles and fields and they change over time and contribute to the total forces and torques that objects feel). They start what they are and they change over time.
A: https://physics.stackexchange.com/a/65392/101895
Brilliantly explained there, relate the contracting effect with classical sound Doppler effect if you could. Again, only force acting on a particle due to another particle is solely coulomb's force.The most important thing is that concept of exchange of particles is supposed to create the force {basically, a result to make calculations easier}, i.e. photons, their speed being c, just to understand it logically, suppose we leave out relativity and assume that c is any normal speed, then the "density" {number of photons effectively emmited in a particular direction in unit time} of photon being emitted would depend on speed of the particle due to which force need to be calculated at the time of photon emmition and again, the acceptor or the particle on which force need to be calculated , its speed would also effect the momentum imparted by photon to it at the time of photon acception, Hence a device was designed keeping relativity in mind too. 

if the acceptor was on right, it would recieve lots more photons and force than if it was on left.
Also, as evident, magnetic forces do no work, as their sole job is to "fix" the instantaneous net force and they have no potential
