Direction of magnetic force Does magnetic force act along the line

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.
$\overrightarrow{F}=q_2(-\mathrm{\Delta }\overrightarrow{v}\times \overrightarrow{B})$
$\overrightarrow{B}=(\overrightarrow{\mathrm{\Delta }}v\times \overrightarrow{\mathrm{\Delta }}x)(\frac{q_1}{||\mathrm{\Delta }\overrightarrow{x}|{|}^3}\frac{{\mu }_0}{4\pi })$
And got this for the answer by direct subtitution:
$\overrightarrow{F}=q_2(-\mathrm{\Delta }\overrightarrow{v}\times (\overrightarrow{\mathrm{\Delta }}v\times \overrightarrow{\mathrm{\Delta }}x)(\frac{q_1}{||\mathrm{\Delta }\overrightarrow{x}|{|}^3}\frac{{\mu }_0}{4\pi }))$
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 $\mathrm{\Delta }v<<c$? If this equation proves to be wrong, what would be the correct approach?

Magnetic force on capacitor
I have this problem: I have a capacitor formed by two parallel square-shaped plates, of side $a$, and with distance $d$ between them. They're asking me out of other things, the magnetic force as a function of some things. I calculate de magnetic force by calculating the gradient of the magnetic energy, given by:
$E=\frac{1}{2}{\iiint }_{V}B\cdot HdV$
I have calculated $B$ and $H$, that form, during the charge of the capacitor, loops around the center proportional to $r$, being $r$ the distance from the central axis. The problem is that I have to integrate that to a cubic volume: $a^{2}d$, one of those variables is independent, but the other two are not, and this problems never involve complicated integrals, such that one:
${\int }_0^a{\int }_0^{a}rdxdy$
If the plates were circles it would be trivial, but being squares, I'm sure I'm missing something.

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?

Mysterious magnetic force!
If you have a loop in which current goes clockwise as seen from top, then it forms a north pole if seen from below. Now if a particle with positive charge goes from left to right under it, it experiences and upward force, now if the 1st loop was very large and particle goes from near about the axis of loop, it goes in a circle in anti-clockwise direction as seen from the top. This again generates a north pole if seen from top.
Q1. Since we have two north poles facing each other, why don't we get a new repulsive magnetic force?
If you closely observe this systen then the negative charges travelling in 1st loop in anti-clockwise direction have synchronized with the positive charge going anti-clockwise below it. Not only this, if you send the particle from right to left, it experiences a downward force and again synchronizes, now again if you try sending negative particles they get synchronized in the same way with the positive particles.
Q2. Why is such a synchronizing taking place?
I am more interested in the first question for now, as I am trying to work out the second problem myself.

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 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
$\frac{Uo{I}_1{I}_2}{2\pi d}l=mg$
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?

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 $U=-\overrightarrow{\mu }\cdot \overrightarrow{B}$ which suggests magnetic field can store energy and hence do some kind of work.