As a beginner, I am struggling a bit with CW-complexes. I'm reading Hatcher, chapter 0. So I want to pose a few questions that are almost embarrassing to me but I believe it is important to ask such questions.
1) As is usual, Hatcher defines the $n$-skeleton as $\left(X^{n-1} \bigsqcup_{\alpha} D_{\alpha}^n \right)/\sim$ with the usual equivalence relation. He later says that as a set we have $$X^n = X^{n-1} \bigsqcup_{\alpha} e_{\alpha}^n $$ where $e_\alpha^n$ is an open $n$-disc.
Does he mean that points on the boundaries of $D_{\alpha}^n$ are identified with points in $X^{n-1}$ and so we can think of these points as points on $X^{n-1}$? So what remains of $D_{\alpha}^n$ are the interiors, i.e. $e_{\alpha}^n$...
(This agrees with the pictorial intuition of gluing.)
2) Why do we define the characteristic map $\Phi_{\alpha} : D_{\alpha}^n \to X$? Where is this used? Maybe is better to first ask the more naive question: What does this map even do intuitively/geometrically?
3) A subcomplex $A$ is a closed subspace of a CW-complex $X$ that is an union of cells in $X$. Hatcher says that the image of characteristic functions of cells in $A$ are again in $A$ because $A$ is closed.
Why do we need closedness here?
Regarding your question 1), Hatcher says, on page 5 item (2):
So, as you say, it really is just an issue of retelling the definition: points on the boundaries of $D^n_\alpha$, namely the points $x \in \partial D^n_\alpha = S^{n-1}$, are identified with points in $X^{n-1}$, namely $x$ is identified with $\phi^n_\alpha(x) \in X^{n-1}$. One could also write $\partial D^n_\alpha = S^{n-1}_\alpha$ to resolve a little bit of ambiguity.
Regarding your question 2), you will find, as you read on, that the characteristic maps are used in various important places. They are of chief importance in discussions of homology later on. If you have an on-line version of the book and search for "characteristic map", you can find some of those places, for example: starting on page 105 at the beginning of the discussion of simplicial homology; starting on page 128 when proving the equivalence of simplicial homology and singular homology; on page 141 when proving equivalence of CW and singular homology.
From an intuitive point of view, the characteristic maps tell you how the open cells are related to the rest of the complex. An open $n$-cell $e^n_\alpha$ is not just floating somewhere disconnected from the $n-1$ skeleton. Instead, one can say intuitively that $e^n_\alpha$ is bonded to the $n-1$ skeleton by the characteristic map (see Hatcher's discussion of characteristic maps on page 7). One say also say rigorously that the closure in $X$ of the open cell $e^n_\alpha$ is equal to $\Phi^n_\alpha(D^n_\alpha) = e^n_\alpha \cup \phi^n_\alpha(S^{n-1}_\alpha)$, which is a consequence of the fact that $\Phi^n_\alpha : D^n_\alpha \to X$ is continuous and that the closure in $D^n_\alpha$ of its interior $\text{int}(D^n_\alpha)$ is equal to $D^n_\alpha = \text{int}D^n_\alpha \cup S^{n-1}_\alpha$.
Regarding your question 3), since $\Phi^n_\alpha(D^n_\alpha)$ is the closure of $e^n_\alpha$, it follows that if a closed subset $A \subset X$ contains $e^n_\alpha$ then $A$ contains $\Phi^n_\alpha(D^n_\alpha)$.