Table Row Color Covers Text

The white space comes from booktabs adding some vertical padding above and below table rules. A workaround consists in setting this padding to $0$ pt and replacing it with vertical spacing with the cellspace package: it defines a minimal such vertical spacing and is compatible with \rowcolors (the column specifier must be preceded by the letter S).

As for the issue with @{}, I replace it with adding negative horizontal spacing on leaving cells in the second column and on entering the third, with >{} and <{}.

\documentclass[preview]{article}
\usepackage{mathtools}
\usepackage{tabularx, booktabs, caption, array}
\usepackage{colortbl}
\usepackage[table]{xcolor}
\usepackage{cellspace}
\setlength\cellspacetoplimit{6pt}
\setlength\cellspacebottomlimit{6pt}

    \begin{document}

\begin{table}
\setlength\aboverulesep{0pt}
\setlength\belowrulesep{0pt}
 \rowcolors{2}{gray!15}{white}% <{\hskip-\arraycolsep}>{\hskip-\arraycolsep{}}
    $ \begin{array}{Slr <{{}\hskip-\arraycolsep}>{\hskip-\arraycolsep\mkern-.5mu}lcc}
        && & \text{Col 1} & \text{Col 2} \\
        \toprule
        \text{Row 1} & (100 \times{} & 100) & 1 & 2 \\
        \text{Row 2} & (100 \times{} &1000) & 3 & 4 \\
        \bottomrule
    \end{array} $
\end{table}

\end{document} 

For the first question, you can remove the \tabcolsep separately for those two columns like

r<{\hspace{-\tabcolsep}}>{\hspace{-\tabcolsep}\,}c
                    <{\hspace{-\tabcolsep}\,}>{\hspace{-\tabcolsep}}lcc}

and for second, you can define a \bottomrulec like

\newcommand{\bottomrulec}{%
  \arrayrulecolor{gray!15}\specialrule{\belowrulesep}{0pt}{0pt}
  \arrayrulecolor{black}\specialrule{\heavyrulewidth}{0pt}{0pt}
  \arrayrulecolor{black}
}

and use it instead of \bottomrule. Here are the coloured versions for \toprule and \midrule in case if you need them.

\newcommand{\toprulec}{%
  \arrayrulecolor{black}\specialrule{\heavyrulewidth}{\aboverulesep}{0pt}
  \arrayrulecolor{gray!15}\specialrule{\belowrulesep}{0pt}{0pt}
  \arrayrulecolor{black}
}
\newcommand{\midrulec}{%
  \arrayrulecolor{gray!15}\specialrule{\aboverulesep}{0pt}{0pt}
  \arrayrulecolor{black}\specialrule{\lightrulewidth}{0pt}{\belowrulesep}
}

Your code modified:

\documentclass{article}
\usepackage[table]{xcolor}
\usepackage{booktabs}
\newcommand{\bottomrulec}{% Coloured \toprule
  \arrayrulecolor{gray!15}\specialrule{\belowrulesep}{0pt}{0pt}
  \arrayrulecolor{black}\specialrule{\heavyrulewidth}{0pt}{0pt}
  \arrayrulecolor{black}
}

\begin{document}

\begin{table}
    \footnotesize
    \centering
    \rowcolors{2}{gray!15}{white}
    \begin{tabular}{lr<{\hspace{-\tabcolsep}}>{\hspace{-\tabcolsep}\,}c
                        <{\hspace{-\tabcolsep}\,}>{\hspace{-\tabcolsep}}lcc}
        &&&& Col 1 & Col 2 \\ \toprule
        Row 1 & $(100$ & $\times$ & $100)$    & $1$ & $2$ \\
        Row 2 & $(100$ & $\times$ & $1000)$& $3$ & $4$ \\
     \bottomrulec
\end{tabular}
\end{table}

\end{document}