How does a solar wind plasmoid split when it touches the magnetosphere?

The plasma travels pretty fast towards earth (light speed?). When it hits the bow shock, it gets slowed down.

No, plasma does not move at light speed. It does flow from the sun at $geq$300 km/s (typically closer to 400 km/s and upwards of 800 km/s). Yes, the plasma that intersects with the bow shock is decelerated, as all flows are passing through a shock wave. The plasma that misses the bow shock, however, just keeps on flowing at its previous speed.

And a short time after it touches the magnetosphere. What will happen next depends if the plasma is a solar wind or a CME?

Kind of, yes. The primary difference is that coronal mass ejections (CMEs) tend to have higher ram pressure and magnetic field geometries conducive to affecting the Earth's magnetosphere.

CME case: The plasma has still such a high velocity that a magnetic reconnection happens. So the outer field lines of the magnetosphere are splitting at the magnetic equator and are folding upwards/downards towards the earth's poles. (I'm looking from the side of the earth. So the solar wind comes from left/right)

No, not really. The speed of the CME does not determine whether magnetic reconnection occurs. The geometry of the magnetic field in the CME vs the magnetosphere determine the rate of reconnection. As for what happens in reconnection, I wrote a an answer at https://physics.stackexchange.com/a/559759/59023 that may be useful background.

Solar wind case: The ions of the plasma travel along the outer field lines of the magnetosphere in a spinning fashion. (Due to Lorentz force? Or by what influence is it possible that the electrons can travel upwards AND downwards along the field lines which are vectors that are solely pointing upwards to the earth's magnetic south pole?)

I am really not sure what you are trying to describe here. I could guess you are thinking about single particle trajectories in a dipole magnetic field. If so, look up topics on trapped particles in the radiation belts to see the three stable drifts of particles in a dipole geometry.

And yes, all of this is always due to the Lorentz force. Particles move along the magnetic field because the part of the Lorentz force associated with the magnetic field acts orthogonally to the magnetic field. That is, in the absence of electric fields the particles can stream exactly along a magnetic field without experiencing a force so long as the field is homogeneous/uniform and any changes occur much more slowly than a typical gyroperiod.

As for resources imaging, you may want to look up Goddard's Science Visualization Studio (SVS).