What mechanism causes oscillations of the solar system's orbit about the galactic plane?
The cause for the oscillations perpendicular to the galactic plane is the gravity of the non-spherical mass distribution (needed for a plane Kepler ellipse) in the Milky Way. Simplified, there is a dense galactic plane. The density is not exactly known; therefore there is some uncertainty (a few million years) about the precise oscillation period. Details see this article, subsection 3.3.

The idea of a correlation of mass extinctions with this oscillation isn't new, it originates back probably to about 1970, or earlier.

"The outer solar system probably does not contain a large gas giant planet, or a small, companion star", see this press release.

We are probably not orbiting around a spiral arm.

The dark matter disk is a hypotheses, an idea to investigate. Usually only a small fraction of this kind of hypotheses can be confirmed definitively later, most of them can be ruled out after some time, some remain unsolved, some can be refined to match observations.