Add another color at other keywords in listings

Since you want to customize (La)TeX, it's a good idea to set texcsstyle to have the same color as the keywords and to use the star so that the backslash will also have the same color; regarding your issue, the braces are not really keywords; I set their style (and also for the square brackets) using literate (notice the * character in the first literate line to prevent color form appearing in comments and strings):

\documentclass{scrreprt}
\usepackage{listings}
\usepackage[dvipsnames]{xcolor}

\definecolor{keywords}{HTML}{8A4A0B}
\definecolor{background}{HTML}{EEEEEE}
\definecolor{comments}{HTML}{868686}
\definecolor{myblue}{RGB}{20,105,176}

\lstset{language=[LaTeX]Tex,
    keywordstyle=\color{keywords},
    texcsstyle=*\color{keywords},
    basicstyle=\normalfont\ttfamily,
    commentstyle=\color{comments}\ttfamily,
    stringstyle=\rmfamily,
    numbers=left,
    numberstyle=\scriptsize,
    stepnumber=1,
    numbersep=8pt,
    showstringspaces=false,
    breaklines=true,
    frameround=ftff,
    frame=lines,
    morekeywords={RequirePackage,ProvidesPackage,NeedsTeXFormat},
    backgroundcolor=\color{background},
    literate=
            *{\{}{{\textcolor{myblue}{\{}}}{1}
            {\}}{{\textcolor{myblue}{\}}}}{1}
            {[}{{\textcolor{myblue}{[}}}{1}
            {]}{{\textcolor{myblue}{]}}}{1},
}

\begin{document}

\begin{lstlisting}[firstnumber=1]
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{arsclassica}[2012/02/21 v4.0 Customizing ClassicThesis (LP)]
\RequirePackage{classicthesis}
\RequirePackage{caption}% Caption package
\end{lstlisting}

\end{document}

To answer your specific question, you can use [<number>] both in keywordstyle and in keywords, to define different keyword sets with their own style; a little example

\documentclass{scrreprt}
\usepackage{listings}
\usepackage[dvipsnames]{xcolor}

\definecolor{keywords}{HTML}{8A4A0B}

\definecolor{background}{HTML}{EEEEEE}

\lstset{language=[LaTeX]Tex,
    keywordstyle=\color{keywords},
    keywordstyle=[2]\color{red},
    keywordstyle=[3]\color{orange},
    basicstyle=\normalfont\ttfamily,
    showstringspaces=false,
    frameround=ftff,
    frame=lines,
    morekeywords={RequirePackage,ProvidesPackage},
    keywords=[2]{Some,Other,Keywords},
    keywords=[3]{and,another,test},
    backgroundcolor=\color{background}
}

\begin{document}

\begin{lstlisting}[firstnumber=1]
\NeedsTeXFormat{LaTeX2e}
Some Other Keywords
and another test
\end{lstlisting}

\end{document}

To apply color to lengths, one can escape to LaTeX (this however won't work for in-line listings):

\documentclass{scrreprt}
\usepackage{listings}
\usepackage[dvipsnames]{xcolor}

\definecolor{keywords}{HTML}{8A4A0B}
\definecolor{background}{HTML}{EEEEEE}
\definecolor{comments}{HTML}{868686}
\definecolor{lengthcolor}{RGB}{200,40,150}
\definecolor{myblue}{RGB}{20,105,176}

\lstset{language=[LaTeX]Tex,
    keywordstyle=\color{keywords},
    texcsstyle=*\color{keywords},
    basicstyle=\normalfont\ttfamily,
    commentstyle=\color{comments}\ttfamily,
    stringstyle=\rmfamily,
    numbers=left,
    numberstyle=\scriptsize,
    stepnumber=1,
    numbersep=8pt,
    showstringspaces=false,
    breaklines=true,
    frameround=ftff,
    frame=lines,
    morekeywords={RequirePackage,ProvidesPackage,NeedsTeXFormat,setlength},
    backgroundcolor=\color{background},
    literate=
            *{\{}{{{\color{myblue}{\{}}}}{1}
            {\}}{{{\color{myblue}{\}}}}}{1}
            {[}{{{\color{myblue}{[}}}}{1}
            {]}{{{\color{myblue}{]}}}}{1},
   escapeinside=!!
}

\newcommand\lencolor[1]{\textcolor{lengthcolor}{\texttt{#1}}}

\begin{document}

\begin{lstlisting}[firstnumber=1]
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{arsclassica}[2012/02/21 v4.0 Customizing ClassicThesis (LP)]
\RequirePackage{classicthesis}
\RequirePackage{caption}% Caption package
\setlength\parskip{!\lencolor{18pt}!}
\hspace*{!\lencolor{15cm}!}
\end{lstlisting}

\end{document}