This at least compiles and produces output. Removed the extra closing brace in \usepackage{breqn}} and the extra \ in \end{\equation}, and commented out the \begin{comment}, \end{comment} blocks. Plus I think you had some extra blank lines that I removed, and I think the quad was intended to be \quad. Hope this gets you going...
\documentclass{article}%
\usepackage{amsmath}%
\usepackage{amsfonts}%
\usepackage{amssymb}%
\usepackage{graphicx}
\usepackage{verbatim}
\usepackage{breqn}
%-------------------------------------------
\begin{document}
%this equation 1 is ok
\begin{equation}
g\left[Pr(happy_{im}=1)\right]=\alpha_{0i}+\alpha_{1}.\left|living\_shock_{im}\right|+\epsilon_{im}
\label{Original}
\end{equation}
%this equation 2 is ok
\begin{equation}
g\left[Pr(happy_{im}=1)\right]=\alpha_{0i}+\alpha_{1}.\left|living\_shock_{im}\right|+\beta.{year}+\epsilon_{im}
\label{yearlytrend}
\end{equation}
%to split3
%have commented it out as no pdf output is produced otherwise
%\begin{comment}
\begin{equation}
\begin{split}
g\left[Pr(happy_{im}=1)\right]=\alpha_{0i}+\alpha_{1}.
\left|living\_shock_{im}\right|+\\
\quad+\beta.{year}+\gamma.region.
\left|living\_shock_{im}\right|+\delta{region}+\epsilon_{im}
\label{region}
\end{split}
\end{equation}
%\end{comment}
%again no output produced when {dmath} environment of breqn used; commented out
%\begin{comment}
%to split4
\begin{dmath}
g\left[Pr(happy_{im}=1)\right]=\alpha_{0i}+\alpha_{1}.\left|
living\_shock_{im}\right|+\beta.{year}+//+\gamma.demo_{iy}.\left|
living\_shock_{im}\right|+\delta_{iy}.demo_{iy}+\epsilon_{im}
\label{demo}
\end{dmath}
%\end{comment}
%to split5
\begin{equation}
\begin{split}
g\left[Pr(happy_{im}=1)\right]=\alpha_{0i}+\alpha_{1}.\left|living\_shock_{im}\right|+\beta.{year}+\gamma_{1}.demo_{iy}.\\
\left|living\_shock_{im}\right|+\gamma_{2}.urbanpop\%_{iy}.\left|living\_shock_{im} \right|+\delta_{1}.demo_{iy}+\delta_{2}.urbanpop\%_{iy}+\delta_{3}.population+\epsilon_{im}
\label{size}
\end{split}
\end{equation}
\end{document}
//did not work. Are you perhaps using\[ \]for this equation in which case you can't use//? – Peter Grill Aug 06 '11 at 17:26\\? – percusse Aug 13 '11 at 19:21