Is the hydrophobic force also known as the Van der Waals bonds? Or are they completely different? Is it that hydrophobic forces keep the water molecules close by? and the actual force between the octane molecules is the van der Waals bonds?
The hydrophobic force squeezes the octanes together, it does nothing TO the water, it is is force created BY the water. By cuddling up to each other, the octanes present a smaller surface area to the surrounding water molecules, and so more water molecules are free to be disorganized. Since disorganization (maximum entropy) is the preferred state, the octane stay together.
Van der Waals bonding is not related to the hydrophobic forces. The van der Waals bonds are created when two atoms come into close proximity, as one atom's electron cloud induces an opposite charge on the other atom's periphery. The octane molecules that are pushed together by hydrophobic forces will interact with each other through van der Waals bonding, just as any two molecules that are closely positioned will do.

When discussing hydrophobic interactions the lecture notes say, "water molecules surrounding an apolar molecule take on a relatively ordered structure." This idea is also pictorially represented by all the water molecules around octane having the same orientation. Why does water naturally do this when it is surrounding "an apolar molecule"?
The picture is not meant to imply a particular orientation for the water molecules surrounding the octane, but just that they are constrained from totally free movement: they can only H-bond to other water molecules on one side, but not on the other. Without the octane, they would have more possibilities, and so a higher entropy.

I don't get how or why water  molecules form a cage structure around nonpolar molecules. I understand the spheres of hydration Becker describes, but those have to do with ions, and in his discussion of the relationship between water and organic molecules (24) he doesn't mention the water cage or entropy.  I don't see why a polar molecule would want anything to do with something like octane in the first place -- much less a structured, ordered relationship.
It seems that the water molecules around the hydrophobic molecules get locked into a more specific arrangement with their neighboring water molecules, forming this cage (or "clathrate") or even ice-like structure in this microenvironment. It's not really ice since it does not last that long. But it's long enough to represent a considerably more ordered structure than the case in which the water molecules form more random partnerships, in the absence of the hydrophobic intruder.  Try these sites for more explanations:
http://www.aw.com/mathews/ch02/clathrat.htm
http://wine1.sb.fsu.edu/bch5887/lecture_6/lecture_6.html