I would intuitively expect that the positive particles will go one way, and the negative particles (electrons) will go the opposite way, as per Lorentz force.
The Lorentz force is acting on the particles in the standard way you would expect, but the magnetic field close to the Earth (i.e., within ~2-4 $R_{E}$) near the geographic equator is roughly orthogonal to the plane defined by the equator (well, within ~$10^{\circ}$ or so due to the dipole tilt angle). So the magnetic part of the Lorentz force, $q \ \mathbf{v} \times \mathbf{B}$, only creates a force in the plane of the magnetic equator orthogonal to the Earth's magnetic field.
If the particles are moving only along the magnetic field, then $q \ \mathbf{v} \times \mathbf{B} = 0$, so there is no resulting charge separation in the sense you implied in your question.
There is a current set up during geomagnetic storms when charged particles are injected from the tail. This results from gradient and curvature drifts that contain sign dependences. The net result is that the electrons drift towards the dawn side while positively charged ions towards the dusk side. Most of the particles will drift according to the three adiabatic invariants: 1st causes gyration about the local field; 2nd causes bouncing between magnetic poles; and 3rd is associated with radial transport (i.e., radially toward/away from Earth due to changes in the background magnetic field strength).
- Separating the positives and the negatives will upset the neutral electric charge.
Generally this does not happen in plasams, as they tend to stay quasi-neutral.
- The solar wind itself is already magnetized, hence, it might not be as straight-forward as "a clod of particles going somewhere uniformly", but rather, some sort of internal turbulence also occurs.
Yes, the solar wind does disrupt a "quiet" magnetosphere, but this generally alters particle drift paths. It would not produce large scale charge separation like your question suggests.
- The strength of the solar wind's magnetic field is not negligible when compared to the Earth's, hence the local magnetic field felt by the particles isn't static.
Well, this is kind of correct. The Earth's quasi-static magnetic field ranges from ~25,000-65,000 nT while the solar wind is generally ~5-10 nT. Thus, the Earth's field is anywhere from ~2500-13,000 times larger than the solar wind's field.
will one pole collect the positive charges, and the other the negatives?
No, this is not how it works as I explained above. I wrote a more detailed explanation at https://physics.stackexchange.com/a/257548/59023 and https://physics.stackexchange.com/a/253491/59023 as to why plasmas generally do not "charge up."
