Evaluate $\int_{0}^{\pi} \frac{d\theta}{(2+\cos\theta)^2}$

I will start from the point where you left off. The integral may be written as, upon substitution of $z=e^{i \theta}$:

$$\begin{align}\frac{1}{2} \int_0^{2 \pi} \frac{d\theta}{(2+\cos{\theta})^2} &= -i 2 \oint_{|z|=1} dz \frac{z}{(z^2+4 z+1)^2}\\ &= -i 2 \oint_{|z|=1} dz \frac{z}{(z+2-\sqrt{3})^2 (z+2+\sqrt{3})^2}\end{align}$$

after a bit of algebra. This integral is $i 2 \pi$ times the sum of the residues of the poles within the integration contour. The only pole inside this contour, as you point out, is the pole at $z=-2+\sqrt{3}$. The other pole at $z=-2-\sqrt{3}$ is outside this contour and is not counted.

To compute the residue at this pole, note that we have double roots. For such roots, we have to take a derivative:

$$\mathrm{Res}_{z=-2+\sqrt{3}} \frac{-i 2 z}{(z+2-\sqrt{3})^2 (z+2+\sqrt{3})^2} = \lim_{z \rightarrow -2+\sqrt{3}} \left [\frac{d}{dz} \frac{-i 2 z}{(z+2+\sqrt{3})^2} \right ]$$

I will leave the algebra to the reader; the result is $-i \sqrt{3}/9$. The integral is then

$$\int_0^{\pi} \frac{d\theta}{(2+\cos{\theta})^2} = i 2 \pi \frac{-i \sqrt{3}}{9} = \frac{2 \sqrt{3}}{9} \pi$$

Trigonometric substitution:

$$x=\tan\frac{\theta}{2}\Longrightarrow d\theta=\frac{2}{x^2+1}dx\;\;,\;\;\cos\theta=\frac{1-x^2}{1+x^2}\Longrightarrow$$

$$\int\limits_0^\pi\frac{d\theta}{(2+\cos\theta)^2}=\int\limits_0^\infty\frac{2\,dx}{1+x^2}\frac{1}{\left(2+\frac{1-x^2}{1+x^2}\right)^2}=2\int\limits_0^\infty\frac{x^2+1}{(x^2+3)^2}dx=$$

$$2\int\limits_0^\infty\left(\frac{1}{3+x^2}-\frac{2}{(3+x^2)^2}\right)=\left.\frac{2}{\sqrt 3}\arctan\frac{x}{\sqrt 3}\right|_0^\infty-\left.4\left(\frac{x}{6(3+x^2)}+\frac{1}{6\sqrt 3}\arctan\frac{x}{\sqrt 3}\right)\right|_0^\infty=$$

$$=\frac{4}{3\sqrt 3}\frac{\pi}{2}=\frac{2\pi}{3\sqrt 3}$$

$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\bbox[10px,#ffd]{\int_{0}^{\pi}{\dd\theta \over \bracks{2 + \cos\pars{\theta}}^{\,2}}} = \left.-\,\totald{}{\mu}\int_{0}^{\pi}{\dd\theta \over \mu + \cos\pars{\theta}} \right\vert_{\ \mu\ =\ 2} \\[5mm] = &\ -\,\totald{}{\mu}\braces{\int_{0}^{\pi/2}{\dd\theta \over \mu + \cos\pars{\theta}} + \int_{-\pi/2}^{0}{\dd\theta \over \mu - \cos\pars{\theta}}}_{\mu\ =\ 2} \\[5mm] = &\ \left.-\,\totald{}{\mu}\int_{0}^{\pi/2}{2\mu \over \mu^{2} - \cos^{2}\pars{\theta}}\,\dd\theta\,\right\vert_{\ \mu\ =\ 2} \\[5mm] = &\ \left.-2\,\totald{}{\mu}\int_{0}^{\pi/2}{\mu \over \mu^{2}\sec^{2}\pars{\theta} - 1} \,\sec^{2}\pars{\theta}\,\dd\theta\,\right\vert_{\ \mu\ =\ 2} \\[5mm] = &\ \left.-2\,\totald{}{\mu}\int_{0}^{\infty}{\mu \over \mu^{2}t^{2} + \mu^{2} - 1} \,\dd t\,\right\vert_{\ \mu\ =\ 2} \\[5mm] = & \left.-\pi\,\totald{}{\mu}{1 \over \root{\mu^{2} - 1}}\right\vert_{\ \mu\ =\ 2} = \bbx{\ds{{2\root{3} \over 9}\,\pi}} \end{align}