LuaTeX-ja

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> Instead, I want gothic to be the default kanji style for the entire document. How do I do this?

Load luatexja-fontspec and use \setmainjfont.
Example:

\documentclass{article}
\usepackage{luatexja-fontspec}
\usepackage{minted}
\setmainjfont{IPAexGothic}
\begin{document}
一二三四五六七八九〇
\begin{minted}[linenos]{python}
print("Hello")
print("こんにちは")
\end{minted}
\end{document}


> I would like all Kanji appearing in the code listings to be gothic format

This is also possible by redefining \ttfamily.
Example:

\documentclass{article}
\usepackage{luatexja}
\makeatletter
\DeclareRobustCommand\ttfamily
{\not@math@alphabet\ttfamily\mathtt
\romanfamily\ttdefault\kanjifamily\gtdefault\selectfont}
\makeatother
\usepackage{minted}
\begin{document}
一二三四五六七八九〇
\begin{minted}[linenos]{python}
print("Hello")
print("こんにちは")
\end{minted}
\end{document}