Hatcher: Proof of Cellular Boundary Formula

I'm confused about a couple of points on the proof given in Hatcher (on page 141).

The first line of the last paragraph reads "The map $\Phi_{\alpha_{*}}$ takes a chosen generator $[D^{n}_{\alpha}]\in H_{n}(D^{n}_{\alpha},\delta D^{n}_{\alpha})$ to a generator of the $\mathbb{Z}$ summand of $H_{n}(X^{n},X^{n-1})$ corresponding to $e^{n}_{\alpha}$," where here $\Phi_{\alpha_{*}}$ is the map induced by the attaching map $H_{n}(D^{n}_{\alpha},\delta D^{n}_{\alpha})\rightarrow H_{n}(X^{n},X^{n-1})$ for some CW complex $X$. Are we taking the genrator $[D^{n}_{\alpha}]$ to be the identity singular simplex $\Delta^{n}\rightarrow D^{n}_{\alpha}$? If so, this seems to suggest that the free generators of $H_{n}(X^{n},X^{n-1})$ are exactly the characteristic maps of the $n$-cells - why is this?

Also, later in the proof he says that the map $q_{\beta_{*}}:\widetilde{H}_{n-1}(X^{n-1}/X^{n-2})\rightarrow\widetilde{H}_{n-1}(S^{n-1}_{\beta})$ induced by collapsing the complement of the $(n-1)$-cell $e^{n-1}_{\beta}$ is projection on to the $\mathbb{Z}$-summand corresponding to $e^{n-1}_{\beta}$, but I think I can see why this is true if my guess about the previous point is correct.

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