Are the orientations of our solar system and others in our galactic disc "in-line" with the disc or are they oriented in all different directions? What determines their orientation?
They're oriented in all different directions. The size of a solar system is so much smaller than the size of the Galaxy, that the Galaxy's structure has no impact on the orientation of a solar system. What determines their orientations is the direction of the angular momentum that the system had when it formed, and that's pretty much random.
Our own solar system is tipped by about 63 degrees with respect to the plane of the galaxy. You can see that on this infrared picture taken by the IRAS satellite. The picture is a little tricky to interpret because, like many maps of the Earth, it's an Aitoff projection, which means that the entire sky has been flattened onto an ellipse. But you should be able to see that the angle between the bright horizontal band (the Milky Way's disk) and the blue haze (dust in the plane of the solar system) crosses at an angle of something like 60 degrees.
This page was last updated June 28, 2015.