Apart from the numbering style (which must be setup differently and that's a different question) the 'contraction' of the reference style can be achieved easily with zref
when links are required, just replace \label
with zlabel
and setup a \fullref
command that extracts the anchor, default value and title of the link.
\documentclass{book}
\usepackage{hyperref}
\usepackage[user,hyperref,titleref]{zref}
\makeatletter
\newcommand{\fullref}[1]{%
% Check for existence and provide a hyperlink
\zref@ifrefundefined{#1}{%
}{%
\hyperlink{\zref@extract{#1}{anchor}}{\zref@extract{#1}{default} \zref@extract{#1}{title}}%
}%
}
\makeatother
\begin{document}
See \fullref{my-label}
\chapter{Example-heading}
\section{Example-subheading}
\section{Example-subheading 2}
\subsection{Example subsubheading}
\subsubsection{Example subsubheading}
\subsubsection{Example subsubheading 2} \zlabel{my-label}
\chapter{Example-heading 2}
\section{Example-subheading again}
\end{document}

Update for the hausarbeit-jura.cls
\documentclass{hausarbeit-jura}
\usepackage{expl3}
\usepackage{hyperref}
\usepackage[counter,user,hyperref,titleref]{zref}
\makeatletter
\zref@newprop{chapterprop}{\thechapter}
\zref@newprop{sectionprop}{\thesection}
\zref@newprop{subsectionprop}{\thesubsection}
\zref@newprop{subsubsectionprop}{\thesubsubsection}
\zref@addprops{main}{chapterprop,sectionprop,subsectionprop,subsubsectionprop}
\ExplSyntaxOn
\cs_new:Npn \extractstructure #1{
\str_case_x:nn {\zref@extract{#1}{counter} }
{
{chapter} {\zref@extract{#1}{chapterprop} }
{section} {\zref@extract{#1}{chapterprop}~\zref@extract{#1}{sectionprop} }
{subsection} {\zref@extract{#1}{chapterprop}~\zref@extract{#1}{sectionprop}~\zref@extract{#1}{subsectionprop} }
{subsubsection} {\zref@extract{#1}{chapterprop}~\zref@extract{#1}{sectionprop}~\zref@extract{#1}{subsectionprop}~\zref@extract{#1}{subsubsectionprop} }
}
}
\ExplSyntaxOff
\newcommand{\fullref}[1]{%
% Check for existence and provide a hyperlink
\zref@ifrefundefined{#1}{%
}{%
\hyperlink{\zref@extract{#1}{anchor}}{\extractstructure{#1} \zref@extract{#1}{title}}
}%
}
\makeatother
\begin{document}
See \fullref{my-label} or \fullref{somechapter}
\chapter{Example-heading}
\section{Example-subheading}
\section{Example-subheading 2}
\subsection{Example subsubheading}
\subsubsection{Example subsubheading}
\subsubsection{Example subsubheading 2} \zlabel{my-label}
\chapter{Example-heading 2} \zlabel{somechapter}
\section{Example-subheading again}
\end{document}

\section
,\subsection
, and\subsubsection
-level headers.b)
back to the first chapter and anotherb)
back to another chapter etc.