Formatting new list of definitions with tocloft

documentclass[11pt]{report}
usepackage{tocloft, xparse}

newcommand{listexamplename}{List of Equations}
newlistof{example}{exp}{listexamplename}
newcommand{example}[1]{%
    refstepcounter{example}%
    parnoindenttextbf{Equation theexample. #1}
    addcontentsline{exp}{example}
    {protectnumberline{thechapter.theexample}#1}par}

DeclareDocumentEnvironment{examplee}{ m m }{%
    equationlabel{#1}}{endequationexample{#2}}

begin{document}
    tableofcontents
    listofexample
    listoftables
    chapter{Introduction}
    begin{examplee}{eq:SOE}{Second Order Equation}
        g(pi_j)= alpha_{jk} alpha_{kg}theta_g + alpha_{jk}xi_k + beta_j
    end{examplee}

end{document}

The following minimal example works for me in terms of the book document class:

documentclass{book}
usepackage{tocloft}% http://ctan.org/pkg/tocloft
usepackage{etoolbox}% http://ctan.org/pkg/etoolbox

newcommand{listdefinitionname}{My list of definitions}
newlistof{definition}{def}{listdefinitionname}
newcommand{definition}[1]{%
  refstepcounter{definition}%
  parnoindenttextbf{The Definition~thedefinition. #1}%
  addcontentsline{def}{figure}
    {protectnumberline{thechapter.thedefinition}#1}par%
}
makeatletter
pretochapter{addtocontents{def}{protectaddvspace{10p@}}}%
makeatother

begin{document}

listoffigures
listofdefinition
chapter{A chapter}
begin{figure}caption{Some figure}end{figure}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}

chapter{A chapter}
begin{figure}caption{Some figure}end{figure}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}
begin{figure}caption{Some figure}end{figure}
definition{Some definition}

end{document}

Regarding your list of questions:

1. I've inserted the definition entries to be type figure rather using

   addcontentsline{def}{figure}
     {protectnumberline{thechapter.thedefinition}#1}par%
   

   That way they are formatted similarly, avoiding any overlap. If you have more definitions and still overlap, you need to change the width of the column.

2. My numbering works. You could also use arabic{chapter}.thedefinition or even arabic{chapter}.arabic{definition} which should strip any inherited counters attached to thechapter or thedefinition.

3. Solved via (1.) above.

I've also used the etoolbox package to add a vertical space between the chapter-separated definitions, similar to that of the LoF and LoT. This was done using pretochapter{<stuff>} which adds <stuff> before chapter. Here stuff is addvspace{10p@} which adds a maximum of 10p@ (10pt) vertically.