How is the Plane of the Solar System oriented to the Sun's motion through space: parallel, perpendicular, or some other angle?
Surprisingly it's quite easy to answer this because we can use the cosmic microwave background as a reference. The CMB gives us an average inertial frame for the universe so our motion relative to it is the closest we can come to defining the Solar System's motion through space.
The CMB is isotropic, but because we are moving relative to it the radiation is blue shifted in our direction of motion and red shifted in the opposite direction. This creates the CMB dipole anisotropy - for more detail see this paper on the Arxiv.
So we just need to find the angle relative to the ecliptic where the CMB is hottest. After much scribbling and head-scratching I gave up and resorted to Google, which immediately found this article (NB it's a PDF) that does the calculation for me. The result is that we are moving at about 10° to the plane of the Solar System.
Chris White points out the question The Galactic Plane that points out the tilt of the ecliptic relative to the Milky Way is 60°. The paper by Kogut et al gives the direction of the CMB dipole maximum relative to the Milky Way, so if you know the galactic longitude of the Solar System you should be able to calculate the angle of the dipole maximum from Earth (i.e. the 10° angle mentioned above). It was after trying and failing to do this I resorted to Google :-)