Magnetic Field of a rotating point charge

Not sure if this should be in quantum section, please move if necessary.

We know that:
1. A stationary particle with charge q is not affected by an external magnetic field. We can assume this particle is not magnetic at all, since it would allign and travel in relation to the magnetic effect. Correct?
2. A charged particle, moving at velocity v, is affected magnetically : F=qv X B
3. A rotating charged sphere is also affected, we can picture charges on the surface of the sphere rotating at a speed "v", so it's similar to above.

My question:
What about a charged particle rotating in one place . i.e not moving, only rotating? It has an electric field, because it has charge, and the field thus rotates too.
Will it now have a magnetic field? Will it be affected by an external magnetic field?

My thinking is that we cannot assume the size of the particle as zero, since it has a charge, and a charge requires a surface or volume to exist in, it doesn't exist in zero space. If we thus imagine the point particle as an infinitely small sphere, can we make some sense out of the math of existing equations? Has anyone come across a solution for this?
I am particularly interested in the potential energy and forces involved.