The following proof is taken from Introduction to Topological Manifolds by John Lee
I believe there is a (slight) error in the proof, where the author says "it is the disjoint union of its cells". Shouldn't it just be the union of its cells and not the disjoint union of its cells?
Furthermore to show that $Y$ satisfies the (W) condition for a CW-Complex, one needs to show that topology of $Y$ is coherent with the family of subspaces $\Gamma =\{ \bar{e} | e \in \xi\}$, which means that $U \subseteq Y$ is open in $Y$ if and only if the intersection of $U$ with $\bar{e} \in \Gamma$ is open in $\bar{e}$, however only one direction is proved here.
About your first question: Alex Provost's answer is correct -- the definition of a CW complex guarantees that the cells are disjoint. But maybe you're being misled by the use of the term "disjoint union" -- all I meant here was that $Y$ is equal to a union of cells, and they are disjoint. This is a common informal usage of the term "disjoint union" (described on page 385 of my book), which is different from the abstract disjoint union defined on page 394.
About your second question: Since $Y$ has the subspace topology, any closed subset $S\subseteq Y$ automatically has the property that $S\cap \overline e$ is closed in $\overline e$ for each cell $e\subseteq Y$. So to prove its topology is coherent with the closures of the cells, only the converse needs to be proved. (I pointed this out on p. 131, just after I defined coherent topologies.)