Poles and Bode Plots
Bode plot is not a graph that plots the transfer function (\$H(s)\$) against \$s\$. \$H(s)\$ is a complex function and its magnitude plot actually represents a surface in Cartesian coordinate system. And this surface will have peaks going to infinity at each poles as shown in figure:



Bode plot is obtained by first substituting \$s= j\omega\$ in \$H(s)\$ and then representing it in polar form \$H(j\omega) = |H(\omega)|\angle\phi(\omega)\$. \$H(\omega)\$ gives the magnitude bode plot and \$\phi(\omega)\$ gives the phase bode plot.

Bode magnitude plot is the asymptotic approximation of the magnitude of transfer function (\$|H(\omega)|\$) vs logarithm of frequency in radians/sec (\$\log_{10}|\omega|\$) with \$|H(s)|\$ (expressed in dB) on y-axis and \$\log_{10}|\omega|\$ on x-axis.

Coming to the questions:


At poles, the complex surface of \$|H(s)|\$ peaks to infinity not \$|H(\omega)|\$.
When a system is fed with pole frequency, the cosponsoring output will be having the same frequency but amplitude and phase will be changing. The value can be determined by substituting the frequency in radians/sec in \$|H(\omega)|\$ and \$\phi(\omega)\$ respectively. 
A pole at -2 rad/sec and 2 rad/sec have the same effect on \$|H(\omega)|\$. And our interest is in the frequency response. So we need only positive part of it.