Value of the Riemann Zeta Function for negative integers

Question

I am reading the book "Riemann's Zeta Function" by H. M. Edwards. I had a confusion in Section 1.5, Page 12 at the derivation of the formula of Riemann Zeta Function at negative integers i.e. $\zeta(-n)$ in terms of the Bernoulli numbers.

In the first step the power series expansion $$\dfrac{x}{e^x-1}=\sum_{n=0}^\infty\dfrac{B_nx^n}{n!}$$ where $B_n$ denotes the Bernoulli Numbers, has been substituted into the formula $$\zeta(s)=\dfrac{\Pi(-s)}{2\pi i}\int^{+\infty}_{+\infty}\dfrac{(-x)^s}{e^x-1}\dfrac{dx}{x}$$

Here $\Pi(x)=\Gamma(x+1)$.

I am not sure

1. Where the $|x|=\delta$ comes from?
2. How the variable is changed from $x$ to $\theta$?
3. How the final integral and summation is solved?

I noticed that there was already a question on this Contour Integration, Riemann Zeta (-n). But, this fails to clarify my doubts.