Is it necessary to build the chips using NAND? So far, I've built my way up from the NOT chip to the MUX chip, and the only time I explicitly used NAND was for building NOT, AND and OR. Of course, whenever I use these, I am implicitly using NAND.
Is this fair game? It would seem so to me, but perhaps the book was implying that all the chips in this chapter should be built explicitly using NAND?
Should I go back and re-do chips such as XOR and MUX to include NAND for the sake of having a better understanding of logic design?
You are doing the correct thing. The idea is to learn abstraction, to use the parts you just built to help you build the next parts. Designing complex parts from only Nands takes a lot of work. See Why we like abstraction.