Finding a CW complex structure on a quotient space, and then its universal cover I am given this quotient of the square:

I want to:

Find a homeomorphic CW complex,
Find its fundamental group,
Find a simply connected covering space

I have seen methods for other quotients of the square like the torus, and $\mathbb{R}P^2$ (for a presentation of the fundamental group), but they don't seem applicable here (in particular for the covering space).
 A: Hint: Cut-and-paste. 
If you cut along the diagonal from upper left to lower right (marking the two cut-edges with, say, triple-arrows), and then glue together the two double-arrows, the resulting quotient description should be far easier to work with. 
To answer your question literally, though: Lehrbuch der Topologie, by Seifert and Threlfall (also in English as "A textbook of topology"), has a nice description of how to do all this for a simplicial complex, although the presentation is rather dated. I think you'll find, if you read it, that in this case the distinction between "simplicial" and "CW" is compeletely irrelevant. 
One more thing: If you know the Seifert/van Kampen theorem, computing the fundamental group is relatively easy: take as your set $U$ a concentric (open) square that's 3/4 as big in each direction. And take as $V$ everything outside a (closed) square that's 1/4 as big in each direction. Then $U \cap V$ is basically a circle, a circle that's contractible in $U$ and looks like $aa^{-1}bb$ in $V$, where $a$ and $b$ are the loops generated by the single-arrow and double arrow edges respectively. From this, you can compute $\pi_1(U \cup V)$. 
