As for now the Quiz extension uses raw HTML tags in code to generate the HTML form code for a quiz.
This should be avoided and instead of using this, the better way would be to use a mustache template using the TemplateParser class.
See also: https://www.mediawiki.org/wiki/Manual:HTML_templates
The Quiz extension was created by a volunteer and is currently installed on several WMF projects, such as Wikiversity.
Install mediawiki locally. It is easiest (often) to use vagrant for this. Enable Quiz as a role.
You can test that the extension is working by adding a sample quiz to your test wiki in wikitext. The quiz syntax is well documented here.
You can find the quiz extension in the vagrant repository by navigating to mediawiki/extensions/Quiz. If you haven't already, you will need to set up git, gerrit, and git-review, and install a gerrit hook to begin submitting commits to the Quiz extension.