First, what you are trying to accomplish can be quite difficult. See warning here:
https://discourse.gohugo.io/t/open-built-website-from-local-folder-without-running-a-server/34193/2
Second, ask yourself if it is really worth the effort.
Third, to fix the links to the home page:
mkdir -p layouts/partials
cp themes/ananke/layouts/partials/i18nlist.html layouts/partials/
cp themes/ananke/layouts/partials/site-navigation.html layouts/partials/
layouts/partials/i18nlist.html
{{ if .IsTranslated }}
<h4>{{ i18n "translations" }}</h4>
<ul class="pl0 mr3">
{{ range .Translations }}
<li class="list f5 f4-ns fw4 dib pr3">
{{ $href := .RelPermalink }}
{{ if .IsHome }}
{{ $href = path.Join .RelPermalink "index.html" }}
{{ end }}
<a class="hover-white no-underline white-90" href="{{ $href }}">{{ .Lang }}</a>
</li>
{{ end}}
</ul>
{{ end }}
layouts/partials/site-navigation.html
<nav class="pv3 ph3 ph4-ns" role="navigation">
<div class="flex-l justify-between items-center center">
<a href="{{ path.Join .Site.Home.RelPermalink "index.html" }}" class="f3 fw2 hover-white no-underline white-90 dib">
{{ with .Site.Params.site_logo }}
<img src="{{ . }}" class="w100 mw5-ns" alt="{{ $.Site.Title }}" />
{{ else }}
{{ .Site.Title }}
{{ end }}
</a>
<div class="flex-l items-center">
{{ partial "i18nlist.html" . }}
{{ if .Site.Menus.main }}
<ul class="pl0 mr3">
{{ range .Site.Menus.main }}
<li class="list f5 f4-ns fw4 dib pr3">
<a class="hover-white no-underline white-90" href="{{ .URL }}" title="{{ i18n "pageTitle" . }}">
{{ .Name }}
</a>
</li>
{{ end }}
</ul>
{{ end }}
{{ partialCached "social-follow.html" . }}
</div>
</div>
</nav>
Finally, you still need to sort out a few image URLs (see 404s in your browser’s dev tools).