Understanding Magnetic Force: Equation Examples Right-Hand Rule

Railgun magnetic force does no work?
Q1) The Lorentz magnetic force on a moving electron is always perpendicular to its motion; so no work is done. Yet, why is work done in railgun operation.
Q2) For every action, there must always be a reaction. The recoil force of the railgun will be transmitted to the structure holding the rails. But what is the actual mechanism that transfers the force of reaction?

Magnetic force between 2 moving charges
If 2 positive charges are moving in the same direction with a velocity v relative to the ground frame, will they experience a magnetic force? I’ve read several times that the velocity in the Lorentz force law is relative to the ground frame, not to the object creating the magnetic field, so I feel like there should be a magnetic force.
Moreover, assuming there is a force, I tried computing the velocity at which the electric and magnetic force cancel each other, and got the speed of light. Is this just a coincidence, or is there any proper reasoning behind this?

Why is magnetic field vector perpendicular to magnetic force vector?
So recently in physics class, we learned about the magnetism right hand rules.
One of them states that the index finger points in the direction of the velocity of a particle, the middle finger points in the direction of the magnetic field, and the thumb points in the direction of the magnetic force.
I'm curious why the magnetic field vector is perpendicular to the magnetic force vector.

Magnetic force between two point charges
I tried to derive the magnetic force between two point-charges for iterative computation. Starting out with Lorentz force and Biot–Savart law for a point charge.
$\stackrel{\to <!--\; \to \; -->}{F}=q_2(-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{B})$
$\stackrel{\to <!--\; \to \; -->}{B}=(\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}v\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}x)(\frac{q_1}{||\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{x}|{|}^3}\frac{{\mathrm{\mu <!--\; \mu \; -->}}_0}{4\mathrm{\pi <!--\; \pi \; -->}})$
And got this for the answer by direct subtitution:
$\stackrel{\to <!--\; \to \; -->}{F}=q_2(-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->(\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}v\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{\mathrm{\Delta <!--\; \Delta \; -->}}x)(\frac{q_1}{||\mathrm{\Delta <!--\; \Delta \; -->}\stackrel{\to <!--\; \to \; -->}{x}|{|}^3}\frac{{\mathrm{\mu <!--\; \mu \; -->}}_0}{4\mathrm{\pi <!--\; \pi \; -->}}))$
It does not seem to be right for several reasons.
1.It seems as if electron would be attracted to the nucleus by both magnetic and electrostatic force. Considering hydrogen atom.
2.It is possible to show that between non-moving particle and a non-moving wire with current in it should exist magnetic force. (Magnetic force acts between charge carriers in the wire and the point-charge. Non-moving particles in the wire do not have influence on the magnetic force)
Where have I gone wrong and how to find the correct expression for the magnetic force between two point-charges? Does the equation hold if $\mathrm{\Delta <!--\; \Delta \; -->}v<<c$? If this equation proves to be wrong, what would be the correct approach?

Magnetic force on open circuit?
Let's say we have a straight horizontal wire and we let it drop inside a magnetic field which will be parallel to the ground(coming out of the screen). Charges inside the wire feel a force due to their movement inside the magnetic field .
Let's say the field is coming out of the screen. Positive charges gather on the left and negative on the right. If we had a loop I would have no trouble with this but during the charges' movement do we consider that we have a current? Therefore leading to a magnetic force opposing the bar's drop or not?

Direction of magnetic force
Does magnetic force act along the line joining the centres like gravitational and electric forces do?
Are the directions of magnetic field and magnetic lines of force the same? I have read that the direction of the field is tangential to the direction of the line of force.

Is magnetic force pseudo?
Magnetic force exist only if charge is moving, so it must be pseudo. Imagine, a positively charged man who has the same speed as electron (charge). So, he doesn't feel any magnetic force as charge is at rest with respect to him. Therefore, he only experience electric force.
However a man who is at rest or has different speed than electron feels a magnetic force
Therefore magnetic force must be pseudo. Pls answer me

Can magnetic force do work?
I have been told numerous times that magnetic force do no work at all but I have some trouble digesting this fact. Now suppose we have two straight wire with some current, they certainly can feel force which may be repulsive or attractive depending upon current direction, can magnetic force do work? We also have magnetic potential energy defined to $U=-<!--\; -\; -->\stackrel{\to <!--\; \to \; -->}{\mathrm{\mu <!--\; \mu \; -->}}\cdot <!--\; \cdot \; -->\stackrel{\to <!--\; \to \; -->}{B}$ which suggests magnetic field can store energy and hence do some kind of work.

Direction of magnetic force on a permanent magnet
How would one calculate the direction of the magnetic force on a permanent magnet? I have read about the poles of ferromagnets aligning in a magnetic field, so would the magnetic force just create a torque on the ferromagnet until it aligns with the field?

Why are the electric force and magnetic force classified as electromagnetism?
I confuse the four kinds of fundamental interactions, so I think the electric force and magnetic force should not be classified as a big class called electromagnetism.
Here is my evidence:
1.The Gauss law of electric force is related to the surface integration but the Ampere's law corresponds with path integration.
2.The electric field can be caused by a single static charge while the magnetic force is caused by a moving charge or two moving infinitesimal current.
3.The electric field line is never closed, but the magnetic field line (except those to infinity) is a closed curve.

Magnetic force between moving charges
Given two infinite parallel charged rods with equal charge density $\mathrm{\lambda <!--\; \lambda \; -->}$. They are moving with same constant velocity $\stackrel{\to <!--\; \to \; -->}{v}$ parallel to the rods. Find the speed $v$ for which the magnetic attraction is equal to the electrostatic repulsion.
Well, I know how to solve this problem: we first find out the magnetic field created by one rod on the other using Biot's and Savart's law, then we use the definition of $\stackrel{\to <!--\; \to \; -->}{B}$ ( $d\stackrel{\to <!--\; \to \; -->}{F}=\stackrel{\to <!--\; \to \; -->}{v}dq\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{B}$ ) to find the magnetic force, then equate magnetic and electrostatic forces to find $v$, which will be greater than or equal to $c$, thus conclude it is impossible for the forces to be equal.
However, one can argue as the following:
We all know that "same laws of physics apply in all inertial frames". With a constant velocity $\stackrel{\to <!--\; \to \; -->}{v}$,the rest frame of the rods is an inertial frame. Therefore, if Biot-Savart law applies in our frame, it has to apply in the rest frame. If so, none of the rods will feel a magnetic field from the other one because their relative speed is zero, and there will be no magnetic force between the rods.
I've seen this question several times before in references, exams, exercise sheets,and in many different forms (parallel planes, beam of electrons ...),but no one ever used this argument.What is the problem in it ? Is it something related to Maxwell's equations or special relativity ? Or what else ?
I know a similar question was asked before, but the answers weren't satisfying. Please provide your answers with necessary mathematics.

Electric Force is to Magnetic Force as Gravitational Force is to ...?
One can no nothing about the magnetic force and yet arrive at it by taking the relativistic effects of a current and a moving charge system into account. I ask whether there exists such an inherent force in case of gravity.\

Magnetic force and work
If the magnetic force does no work on a particle with electric charge, then: How can you influence the motion of the particle? Is there perhaps another example of the work force but do not have a significant effect on the motion of the particle?

How mass (a magnet) affects magnetic force?
I'm here asking this as a student. Magnetic force can be effected by distance, the longer the distance, the weaker, the shorter the stronger. But does mass also effect magnetic force? Does a heavier magnet produce stronger magnetic force, compare to lighter one. when they're measured in the same distance? If so, is there an equation or formula to calculate it, the strength of the magnetic force? I've been looking for calculation formulas about magnetic forces for magnets on google, but I can't find anything about how mass affects magnetic force for a magnet. (Maybe I'm just using the wrong key word.)
Ps: This has nothing to do about electromagnetism, just normal magnets.

Which magnetic force is stronger? Electron or proton?
Assume the particles generates the magnetic field with their own spin. No motions except spin.(they have their intrinsic angular momentum.) Then which one generates stronger magnetic force?
I've seen the formula which represents the magnetic moment of the 1/2 spin particles with charge $q$, mass $m$, $\stackrel{\to <!--\; \to \; -->}{\mathrm{\mu <!--\; \mu \; -->}}=\frac{g_{s}q}{2m}\stackrel{\to <!--\; \to \; -->}{S}$ where $g_s$ called g-factor.
But I can't understand the formula.
Let me ask a question, which magnetic force is stronger between electron and proton?

Magnetic force in relation to velocity
I'm misunderstanding something very important regarding magnetic force and its relation to velocity.
According to the Lorentz force, the magnetic force ${\mathbf{F}}_{\mathbf{B}}=q\mathbf{v}\times <!--\; \times \; -->\mathbf{B}$. Assuming the charge is not moving, then $\mathbf{v}=\left(\begin{array}{ccc}0& 0& 0\end{array}\right)$. Therefore, ${\mathbf{F}}_{\mathbf{B}}=0$
So why when I hold a magnet close to a piece of metal, I can feel the magnet applying a force on the piece of metal? Since the piece of metal and the magnet are not moving, shouldn't the net magnetic force be 0? I assume the force I am feeling is from the magnetic field from the magnet and the charges in the piece of metal.

Magnetic force on a stationary body
A charged particle near a current-carrying wire does not experience a magnetic force when its velocity is equal to $0$. So why does a compass needle kept near a current carrying wire experience a force when the compass needle is at rest?

Question about magnetic force
I have some confusion regarding the magnetic force. I know that the magnetic field created by a moving charge or current EXERTS a force on any moving charge or current that is present in the field. But when trying to understand the motion of a charged particle in an uniform magnetic field, the youtube video I saw explained it like this: "The magnetic force on a charged particle ALWAYS points perpendicularly with respect to the velocity and magnetic field. Whenever a force acts on an object perpendicular to its motion, the object will undergo circular motion--this creates a centripetal acceleration)"
I am confused. Is the charged particle exerting a force on itself? Or what is the force that acts on the charged particle that is moving? If the charge particle creates a force due to the magnetic field, is the force it creates itself the force that makes it undergo a circular motion?

Magnetic Force Confusing Paradox
Actually I had posted a very similar question, but I wasn't quite satisfied with the answer so I am posting a new variety again.
Imagine an infinitely long wire carrying current $I_1$ from West to East. At a small distance $d$ above the wire there is another small current carrying wire of length $l$ carrying current $I_2$ from East to West (opposite to the direction of current in the below placed wire). Obviously they are magnetically repelling. And if the second wire (carrying current from East to West) is at rest the magnetic force must be equal to $mg$
The magnetic force is upward and $mg$ is down. The magnetic force can be written as
$\frac{Uo{I}_1{I}_2}{2\mathrm{\pi <!--\; \pi \; -->}d}l=mg$
Now, if the magnetic force is greater than $mg$, the wire moves up. Now magnetic force is up and displacement is up too which means that work done by magnetic force should be positive. How is that possible when we know that work done by a magnetic force is always zero? Is it that an e.m.f. is induced which opposes the change?

Work done by the Magnetic Force
The magnetic part of the Lorentz force acts perpendicular to the charge's velocity, and consequently does zero work on it. Can we extrapolate this statement to say that such a nature of the force essentially makes its corresponding work independent of the choice of path, and hence that the magnetic force is conservative?