How does the carbon chain length affect the polarity of an alcohol? What effect does this have on the solubility of the alcohol in a polar vs. a nonpolar solvent?

Alchols are slightly polar molecules due to their hydroxyl group. If you increase the length of the carbon chain, you are actually increasing the nonpolar domain of the molecule. the C-H bonds that make up the carbon chain are nonpolar. Therefore, as you add more, you essentially "dilute" the polarity of the hydroxyl group and are left with a more and more nonpolar molecule, which becomes less soluble.