Questions about Fourier expansion of an unknown function

Question

I'm quite rusty at Fourier series so you'll have to forgive me if I ask some silly questions. In a book I have, Theory of elasticity by Timoshenko & Goodier, the authors perform a Fourier expansion that I can't quite understand. It can be found on page 289 of the PDF file, 275 of the book. They are trying to solve the equation $$\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=-\frac{q}{S}$$ where $z(x,y)$ and $q,S$ are constants. The area of interest is a rectangle with size $-a\le x\le a$ and $-b\le y\le b$. From what I understand the boundary conditions are that $z(\pm a,y)=z(x,\pm b)=0$.

They then state that "The condition of symmetry wrt the $y$-axis and the boundary conditions at the sides $x=\pm a$ of the rectangle are satisfied by taking $z$ in the form of a series, $$z=\sum_{n=1,3,5,...}^{\infty}b_n \cos\left(\frac{n\pi x}{2a}\right)Y_n$$ in which $b_1,b_3,...$ are constant coefficients and $Y_1,Y_3,...$ are functions of $y$ only."

They do something similar with $-\frac{q}{S}$ where they expand it as $$-\frac{q}{S}=-\sum_{n=1,3,5,...}^{\infty}\frac{q}{S} \frac{4}{n\pi} (-1)^{\frac{n-1}{2}} \cos\left(\frac{n\pi x}{2a}\right)$$ between $-a$ and $a$.

As previously mentioned, I'm a bit rusty when it comes to Fourier expansion so bear with me. I have some questions about these two expansions:

1. Why is $n=1,3,5,...$ and not $n=1,2,3,4,...$?
2. How come $\cos$ was used? When $x=a$ then $\cos\left(\frac{n\pi }{2}\right)=0$ but can't you also expand the equations as $\sin\left(\frac{m\pi }{2}\right)$ where $m=0,2,4,...$? The only reason that I could think of why they didn't do it is that $\cos$ is symmetrical about the $y$ axis.
3. Related to that, is it possible to just modify the contents of the trig terms in the expansion to suit one's purpose as long as they are consistent with the boundary conditions?
4. Where does the $\frac{4}{n\pi} (-1)^{\frac{n-1}{2}}$ come from? I understand that $(-1)^{\frac{n-1}{2}}$ continuously flips the sign but I'm not sure why it's needed. Is it to stop the series just becoming infinitely large or infinitesimal? Is $\frac{4}{n\pi}$ the result of the integral for the calculation of the coefficients?
5. Finally, I think that there's no $0$th coefficient because that would shift the entire series and the boundary conditions would not be respected anymore. Is that correct?