How to get rid of space between section and equation

Use \noindent before \begin{equation}:

\section{Section}
\noindent\begin{equation*}
q = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{equation*}

The reason: TeX operates in vertical mode after \section. If display math starts in vertical mode, the horizontal mode is opened immediately and the line with \parindent is preceded. But you can avoid this line when there is no \parindent. This can be done by \noindent.

You can insert \noindent before each \begin{equation}: if the horizontal mode is opened already the \noindent does nothing. You can create your macro which generates \noindent\begin{equation} or re-define \begin{equation}.

For a cheatsheet, I believe it's better to redefine \section and \subsection.

\documentclass[8pt]{extarticle}
\usepackage[paper=a5paper]{geometry}

\usepackage[english]{babel}

\usepackage{mathtools}
\usepackage{amssymb}

\renewcommand{\section}[1]{%
  \par
  \stepcounter{section}%
  \everypar{\textbf{\thesection\ #1}\quad\everypar{}}%
}
\renewcommand{\subsection}[1]{%
  \par
  \stepcounter{subsection}%
  \everypar{\textbf{\thesubsection\ #1}\quad\everypar{}}%
}

\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\setlength{\lineskip}{0pt}

\AtBeginDocument{%
  \setlength\abovedisplayskip{0pt}%
  \setlength\belowdisplayskip{0pt}%
  \setlength\abovedisplayshortskip{0pt}%
  \setlength\belowdisplayshortskip{0pt}%
}

\begin{document}

\section{Section}
\begin{equation*}
q = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{equation*}

\section{Section}
Some text for explaining the following equation
\begin{equation*}
q = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{equation*}
More text follows the equation

\end{document}

Two ways to solve your problem:

\documentclass[8pt]{extarticle}
\usepackage[paper=a5paper]{geometry}
\usepackage[english]{babel}
%\usepackage{parskip}
%\setlength{\parindent}{0pt}
\usepackage{mathtools}
\usepackage{amssymb}

\usepackage{titlesec}
\titlespacing{\section}{0pt}{0pt}{0pt}
\titlespacing{\subsection}{0pt}{0pt}{0pt}
\titlespacing{\subsubsection}{0pt}{0pt}{0pt}


\def\mycommand{\setlength{\abovedisplayskip}{0pt}%
\setlength{\belowdisplayskip}{0pt}%
\setlength{\abovedisplayshortskip}{0pt}%
\setlength{\belowdisplayshortskip}{0pt}}

\AtBeginDocument{\mycommand}

\let\oldnormalsize\selectfont
\def\selectfont{\oldnormalsize\mycommand}

\newenvironment{myeq}{\vspace{-10pt}\par\begin{equation*}}{\end{equation*}\vspace{-10pt}}

\newsavebox{\mybox}
\newlength{\mylength}
\def\myinleq#1{\savebox\mybox{$#1$}\settowidth{\mylength}{\usebox{\mybox}}\hspace{0.5\dimexpr\linewidth-\mylength-2\parindent\relax\relax}$#1$}

\begin{document}
\section{Section}
\begin{myeq}
q = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{myeq}

\begin{myeq}
q_2 = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{myeq}%emptyline needed here

\myinleq{q_3 = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)}

\section{Section}
\noindent Some text
\begin{equation*}
q = q_0 + q_1i + q_2j + q_3k = (q_0,q_1,q_2,q_3)
\end{equation*}
More text
\end{document}