Does a moving electrically charged particle have a "magnetic charge"? I have feeling that the force on a moving charged particle from an external field is due to the interaction of the external magnetic field with the magnetic field produced by the charged particle. I realize that to understand this better I will likely need a better understanding of relativity, but my question basically boils down to: In a certain reference frame, is the force on a moving charged particle from an external magnetic field due to interactions with the external magnetic field with the magnetic field produced by the particle?
 A: There are two different questions being asked here: one in the title, and one at the end of the body. 
Does a moving electrically charged particle have a "magnetic charge"?
An electric charge produces a divergence in the electric field, which is outlined in Gauss's Law:
$$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$$
Similarly, the existence of a magnetic charge would produce a divergence in the magnetic field. But Maxwell's equations dictate, via Gauss's Law for Magnetism, that
$$\nabla\cdot\mathbf{B}=0$$
no matter what, independently of the configuration or motion of electric charges. As such, a moving electric charge does not acquire a magnetic charge.
Is the force on a moving charged particle from an external magnetic field due to interactions with the external magnetic field with the magnetic field produced by the particle?
In classical electromagnetism, electromagnetic fields from different sources emphatically do not interact. This is why the principle of superposition works - when you calculate the field at a given point, you add up the contributions from each individual source while pretending for a moment that the other sources don't exist. If the fields from different sources interacted, then you wouldn't be able to do this, as you wouldn't be able to consider the field emitted from one source independently of the others. (Note: interference is not the same as interaction. Interference is simply the result of superposition - the amplitude of two propagating waves that interfere at a point will not be affected afterwards by that interference.)
What actually happens is that fields interact with charges. When two electric charges repel, that's because one charge interacts with the field at its location produced by the other charge, and vice versa. When two bar magnets interact, it's because the magnetic field gradient produced by the magnetic dipoles in one magnet produces a force on the magnetic dipoles of the other magnet. And in your example, the moving charge experiences a force because it's interacting with the external fields. Fields don't interact with fields (because superposition works), and charges don't interact with charges (because the motion of a charge does not necessarily immediately lead to a difference in the force felt by another charge, only causing a difference when the change in the field propagates to the other charge). Instead, fields interact with charges. 
A: I don't really understand what you want out of your question, so this may not be an answer, but, what part of:
$$ \vec F = q\vec E + q\vec v \times \vec B$$
do you find insufficient? Moving charges are bent sideways by magnetic fields.
One thing that helped me understand this is the vector potential. When we 1st learn about forces as the gradient of some sort of energy scalar-field, I think it's pretty intuitive that:
$$ \vec E = -\vec{\nabla}\phi $$
Then we learn about a new kind of potential, the vector potential, such that:
$$ \vec B = \vec{\nabla} \times \vec A $$
and it's a little more difficult on the intuition: There's a potential who's curl is a field who interacts with velocities:
$$ \vec F = q\vec v \times (\vec \nabla \times \vec A)$$
what's that all about an why?
The epiphany for me happened when I learned that the Coriolis:
$$ \vec F = 2\vec v \times \vec{\omega} $$
force could be described by a vector potential:
$$ \vec F = \vec v \times (\vec{\nabla} \times \vec u)$$
that is just the velocity, $\vec u(\vec r)$, of your non-internal frame.
That the magnetic field interacting with a charge looks just like fictitious forces acting in a rotating frame hopefully makes you worry less about field-field interactions an gets you more excited about relativity, and why this is the only way it could possibly be.
A: I really liked this MinutePhysics video that talks about the relationship between relativity and electromagnetism. https://www.youtube.com/watch?v=1TKSfAkWWN0
This PBS SpaceTime video goes into more depth and touches on how it relates to quantum invariance https://www.youtube.com/watch?v=V5kgruUjVBs
I can't point you to precise equations but electric and magnetic charge are two sides of the same coin and can't be pulled apart. One is simply the other in a difference reference frame.
This is another good video https://www.youtube.com/watch?v=NqdOyxJZj0U. I actually bought a Lentz tube off of Amazon so I can see this is person. So cool.
A: 
I have feeling that the force on a moving charged particle from an external field is due to the interaction of the external magnetic field with the magnetic field produced by the charged particle. 

Yes. Magnets interact with each other by their magnetic fields.
And if an object has a magnetic field, then it's a magnet.
If an object has a magnetic field in one frame, but not in an other frame, then it's a magnet in one frame but not in an other frame. In which case said object interacts magnetically in one frame, while in an other frame it interacts non-magnetically. Said non-magnetic interaction must be electrostatic interaction - we can know that because of the limited number of interactions that exist in nature.
