Understanding Magnetic Force: Equation Examples Right-Hand Rule

Rate of Acceleration of an Object Pulled by Magnetic Force
Gravity on earth pulls objects toward it with an acceleration of 9.8 m/s/s on Earth until the object reaches it's max potential free fall speed. (I call this terminal velocity though I think that this is wrong by definition)
a) Is there a similar equation for the acceleration of an object being pulled by magnetism?
b) Is there a "terminal velocity" of an object being accelerated by magnetism alone? (This would have to mean that the origin of the magnetic field has zero mass, so I'm thinking this is a theoretical question)
c) Is it possible (however unlikely) that one object can be held in orbit around another by magnetism with minimal gravitational influence? That is to say, can an equilibrium be found between the magnetic force pulling against an object and the velocity of an object. (best example, moon orbiting Earth because of an equilibrium of gravitational force and the velocity of the moon revolving around it)
Thanks in advance for any and all help!
P.S. It would also be interesting to know if there is a difference in the equations for magnetic repulsion, or if the forces are equal.

Give examples of information that in a particular situation is: 1. relevant 2. irrelevant 3. reliable 4. unreliable 5. objective 6. biased 7. complete 8. incomplete 9. useful 10. useless 11. understandable 12. incomprehensible,

Magnetic Force of a current carrying wire loop on a magnetic object
let's say we have a current carrying wire loop WITHOUT AN EXTERNAL MAGNETIC FIELD. It produces a magnetic field on its own. what would the magnetic field and force of this loop on a charge or a magnetic object(or whatever) outside the loop?
I always find the equation of F = IBL which is the force on the on loop from an external magnetic field but i can never find the force that the loop itself produces.
Like i am trying to calculate the force from the loop on an iron ball or find the magnetic field produced by just the loop but outside the loop. If i can get the magnetic field outside the loop then i can derive the force that i need using F = Del(m X B).

How did they arrive at the expression for magnetic force on a charge q?
I understand that the magnetic force exists as a consequence of length contraction between 2 charge carriers in a current carrying wire (wrt the frame of the moving charge); but I don't understand how to arrive at the equation
$F=qvB\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->}).$

Is the magnetic force fully analogous to gravitation?
Superficially, the force between two magnets would seem to be fully analogous to gravity because each is the product of two factors divided by the square of the distance between the objects being attracted.
In other words, if we imagine two celestial bodies with both mass and magnetic poles, it would seem that we could, by adjusting the mass and the magnitude of the magnetic poles, make the forces equal to each other, and as the bodies drew together the two forces would remain equal. In other words, if we were to graph the two forces as a function of distance, the two curves would be identical. Thus, the forces, though they are of a completely different nature, are fully analogous to each other.
Is this correct, or is there some essential difference between the two forces that will always make them behave in a different way?

Can the magnetic force between two magnets be explained classically via magnetization current?
Usually in Electrodynamics courses the magnetic forces are analysed via wire-wire interaction. I don't remember being shown a classical explanation for magnetic forces between magnets. Since it is a quantum system caused mainly by the alignment of spins/magnetic moments, I was wondering if the magnetic force between two magnetized ferromagnets can be explained via Lorentz force on the small magnetization currents formed within the material (even if they are just a mathematical tool in this case).

Magnetic Force on a current carrying conductor
Why does a current carrying conductor parallel to the magnetic field not experience a force?

Why does the magnitude of magnetic force which one moving charge exerts on another charge appears inconsistent in different frames of reference?
I have a fundamental misunderstanding about the definition of magnetic force as arises from Lorenz force: $F=q\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{B}$. According to a basic fact of electromagnetism, electric currents are the source of magnetic fields, and a single moving charge can be viewed as a current. Therefore, in a situation of one moving charge and another stationary charge, the moving charge creates a magnetic field, but by the Lorenz definition of magnetic force, the force experienced by the stationary charge iz zero (since it's velocity is zero). But if we shift to a frame of reference moving at half the speed ($\frac{v}{2}$) of the first charge and in the same direction, we get one charge moving in a speed $+\frac{v}{2}$ and a second charge moving at $-<!--\; -\; -->\frac{v}{2}$. In this scenario, the magnetic force experienced by the same charge is not zero.
So what exactly am i missing? i view it as a serious bug in my fundamental understanding of electromagnetism, and i will be extremely thankful if someone will complete for me the picture of charge-charge electromagnetic interaction.

Why don't we simply define the magnetic field to point on the direction of magnetic force that acts on the moving charge?

Magnetic force between two long conductive wires with current
I was asked to calculate the total magnetic force between two long conductive wires with current on them, flowing in the same direction. I used the formula $F_1={\mu }_o\frac{I_1{I}_{2}l}{2\pi d}$ and got a numerical answer. I then doubled that answer to account for both forces (from either wire to the other). The answer key gives the same number I found, but without doubling it. I do not understand why, because it clearly states in the book that said formula is for the force experienced by wire 1, caused by wire 2, which disregards the force experienced by wire 2, caused by wire 1.

Magnetic force acting on a current carrying wire
I am a high school student and I would like to know why in a magnetic field the Force, $F$, is equal to $BIL\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})$, where $\mathrm{\theta <!--\; \theta \; -->}$ represents the angle between the magnetic field and the current. I understand that $F$ is proportional to $B$, $I$ and $L$, but I do not understand the inclusion of $\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})$. In other words, I would like to see a simple proof or explanation of why the effective length of the wire is $L\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})$

A magnetic field of 37.2 T has been achieved at the MIT Francis Bitter Magnet Labaratory
Find the current needed to achieve such a field near the center of a solenoid with radius 2.60 cm, length 30.0 cm, and 40000 turns.

A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction <0, 3, -4>at the end of the wrench. Find the magnitude of the force needed to supply 100 N-m of torque on the bolt.

What is the correct frame of reference for determining the magnetic force on a charge?
If two charges are both stationary in a given inertial frame, F1, then neither charge should experience a magnetic force due to the presence of the other charge (qv = 0). If we accelerate one charge, but not the other, then again, neither charge should experience a magnetic force, since only one charge has a non-zero velocity as measured in that inertial frame, meaning the other, stationary charge will experience no force in the magnetic field of the moving charge.
Now imagine that we are riding along as part of another inertial frame, F2, and that the first inertial frame discussed above that contains the two electrons F1, is traveling at a relative velocity of v from our perspective (i.e., it’s moving faster than us). Now imagine that we fire two electrons from our inertial frame F2, towards F1, one that travels at a velocity of v from our perspective, thereby traveling at the same velocity as the electron that appeared stationary in F1, and another that travels at the same velocity as the second, "faster" electron.
From our perspective in F2, both electrons have a non-zero velocity: the “slow one” traveling at a velocity of v, and the “fast one” traveling a bit faster than that. From our perspective, in F2, both electrons should experience a magnetic force of attraction due to their non-zero velocities in non-zero magnetic fields, which would change the path of those electrons from the perspective of both inertial frames.
However, from the perspective of F1, the “slow” electron is stationary, and should not experience any magnetic force in any magnetic field.
This seems to not make sense - what would happen as an experimental matter?

Quantifying the magnetic force on a single iron atom
If a single iron atom exists in a non-uniform magnetic field, could one measure the force on it simply by multiplying the magnetic dipole moment of the atom by the change in the magnetic field?

Magnetic force of attracting wires
Two wires with equal current attract each other, but electrons in the wires are moving with same speed (since currents are equal) and are relatively at rest. Then how is the magnetic field produced and the wires attracted?

Magnetic force storage or amplification question
all forms of energy can be stored or amplified for example : capacitor , hydraulics ,laser , pulley , etc..
what is the equivalent to that for magnetic force ? can magnets be connected in series or parallel ?

New "gravity force" analogous to magnetic force?
I was watching Eugene Khutoryansky's physics video about Einstein's Gravito-Electromagnetism, Gravity of moving mass in General Relativity . In that, he discussed why maxwell's electromagnetism laws hold for special relativity also. He took the following situation:
Adam is in a spaceship or the reference frame of two charged particles such that their gravitational force cancels their electromagnetic repulsion. Adam would see no magnetic force since charges are at rest according to his frame. Now Sarah is watching Adam from a ground reference and sees the charges moving and hence they must exert an attractive magnetic force pushing particles towards each other. But now Eugene, said that a force (gravity) analogous to the magnetic force produced by moving charge, is produced by moving mass, which will be repulsive, and will balance the attractive magnetic force.
Now I became confused here, what is this "new gravity force"? I never heard about it before, at least in high school. Is it a mathematical trick or tweaking? Or is there really is a repulsive gravitational force that is exerted by moving masses? So this is my doubt here, can someone please explain this further?