We have an SharePoint Foundation 2013 setup.
At the home page quick launch bar, we have 20 odd subsites, with other subsites off these.
Can I force SharePoint to keep this quick launch bar visible on all sites and subsites?
It is possible for all sub sites.
Go to Site settings >> Navigation. In Current Navigation, select Display the same navigation items as the parent site.
In SharePoint Foundation, you cannot use parent settings for quick launch - only for top navigation bar. You can only select "Use Links from Parent" under top link bar.
SharePoint Server Publishing features not available in SharePoint 2013 Foundation.need to deploy SharePoint Foundation 2013 Publishing Infrastructure solution in SharePoint foundation (more details fallow this link https://spf2013publishing.codeplex.com/) after deploying it, you need to activate SharePoint Foundation Publishing Infrastructure feature from Site Collection Features and Manage site features in SharePoint site's settings. After that you will have NAVIGATION option in your site settings under Look & Feel.