Does the right-hand rule apply to elementary particles, and if not, how can one find the direction of magnetic force of an orbital electron, if the magnetic field, spin, and direction of travel are known?

Yes, the RH rule applies; you still have classical interactions, and a moving charge will feel a force perpendicular to the field and velocity. [math]\vec{F} = q\vec{v}\times \vec{B}[/math]

 

The spin will indicate the magnetic moment, and if you have an inhomogeneous magnetic field, will allow you to see some nonclassical behavior. Orbital electrons will show this nonclassical behavior as well. But this has a different dependence on direction of the magnetic field, since the interaction is [math]\vec{\mu}\cdot{\vec{B}}[/math], so the force will be along the direction of the field gradient.