The first problem is to prove that CW space is compact if and only if it is finite. It's pretty obvious to show that finite CW space is compact because finite collection of cells is compact and so their continuous image under naturalal projection. What I don't understand is the converse. I tried to get the proof from Hatcher's book, but failed. My own approach was to cover the space by open balls which are cells without boundary but they don't cover all the space, so I don't know how to finish the proof this way.
The second problem is to prove that map from CW space is continuous if and only if it's restriction to any subspace ( union of cells of dimension less than some n) is continous.
It's kind of obvious that if this condition doesn't hold then function cannot be continuous and I have a trouble with opposite direction. My suggestion is that any function $f: \coprod_{n}D^n/\sim \to X$ is actually of the form $f: \coprod_{n}D^n \to X$ with $f(x) = f(y)$ iff $ x \sim y$. Is that true?
OK, I've a little more time to answer the question rather than just pass you a reference.
Let $X$ be a compact CW complex and let $E$ be the set of its cells. For each $e\in E$ choose a point $x_e\in e$ and set $A=\{x_e\mid e\in E\}$. Note that $X$ carries the weak topology with respect to the set of its skeletons, and this is equivalent to $X$ carrying the weak topology with respect to its closed cells. Now if $e'$ is any cell in $X$, its closure $\bar{e'}$ meets only a finite number of other cells. Hence if $B\subseteq A$ is any subset then $\bar{e'}\cap B$ is a finite set, and hence closed in $\bar{e'}$ since it is Hausdorff. Since $X$ carries the weak topology with respect to its closed cells we conclude that $B$ is closed in $X$ and therefore also in $A$. We conclude from this that $A$ is discrete and is closed in $X$. However a discrete closed set in a compact space is finite, and this tells us that $E$ is a finite set. That is, $X$ has only a finite number of cells.
For your second question, do you mean subcomplex? Since $X$ carries the weak topology with respect to its skeleta/closed cells this is immediate.