To use (Hg.) after a main editor you can adjust the editortypedelim and editortype field format. (You could do the same for translator.)
To format the titleaddon as you want, you need to patch the title, booktitle, and maintitle macros. I've made a new titleaddondelim to control what is printed. You can also change the titleaddon field format to include the brackets, rather than putting them in your bib file.
MWE
This is taken from your Overleaf document and simplified to make it minimal, so just the problems asked about in this question are dealt with. (In the future, include this kind of code in your question.)
\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{CommissionontheHumanities.1980,
editor = {{Commission on the Humanities}},
year = {1980},
title = {The Humanities in American Life},
subtitle = {Report of the Commission on the Humanities},
location = {Berkeley and London},
publisher = {University of California Press}
}
@book{Doi.1978,
author = {Doi, Yutaka and Sano, Tetsuji and Tanaka, Itsuo},
year = {1978},
title = {Senkyo seido},
titleaddon = {Das Wahlsystem},
volume = {10},
series = {Gendai chih\^o jichi zensh\^u},
location = {T\^oky\^o},
publisher = {Gy\^osei}
}
\end{filecontents}
\usepackage[ngerman]{babel}
\usepackage{csquotes}
\usepackage[style=authoryear]{biblatex}
\addbibresource{\jobname.bib}
\renewcommand{\mkbibnamefamily}[1]{\textsc{#1}}
\DeclareDelimFormat[bib]{nametitledelim}{\addcolon\space}
% Use Hg. in parentheses after editor
\DeclareDelimFormat{editortypedelim}{\addspace}
\DeclareFieldFormat{editortype}{\mkbibparens{#1}}
\DefineBibliographyStrings{german}{%
editor = {Hg\adddot}
}
% Use space instead of period before bracketed titleaddon
\usepackage{xpatch}
\DeclareFieldFormat{titleaddon}{\mkbibbrackets{#1}}
\DeclareDelimFormat{titleaddondelim}{\addspace}
\xpatchbibmacro{title}
{\printfield{titleaddon}}
{\setunit{\printdelim{titleaddondelim}}%
\printfield{titleaddon}}
{}
{}
\xpatchbibmacro{booktitle}
{\printfield{booktitleaddon}}
{\setunit{\printdelim{titleaddondelim}}%
\printfield{booktitleaddon}}
{}
{}
\xpatchbibmacro{maintitle}
{\printfield{maintitleaddon}}
{\setunit{\printdelim{titleaddondelim}}%
\printfield{maintitleaddon}}
{}
{}
\begin{document}
\nocite{*}
\printbibliography
\end{document}

editortypedelimandeditortypeare exactly what I would use (probably I would also redefinetranslatortypeandtranslatortypedelimanalogously). But since the OP mentions Overleaf prominently, I should mention thateditortypedelimwas introduced inbiblatex3.8. but Overleaf runs 3.7. (I would also replace\renewcommand{\labelnamepunct}{\addcolon\space}with\DeclareDelimFormat[bib]{nametitledelim}{\addcolon\space}.) – moewe Nov 06 '18 at 23:56titleaddondelimtobiblatex? I think I've seen a few questions where it makes sense to modify this more easily. – David Purton Nov 07 '18 at 00:10biblatex-extstyles. Not sure what the official stance is on adding all kinds of (useful, no doubt) delimiters to the standard styles. Relevant, though not exactly comparable: https://github.com/plk/biblatex/issues/180 – moewe Nov 07 '18 at 00:21