Is there something in Jetpack that’s also disabling the Home 1-4 widget locations?
No, if you want to have a widgetized front page, you’ll need to configure the theme accordingly (create and set a static front page, place widgets, etc…). Please have a look at the theme documentation for more information.
In the free version of the theme the widget areas on the front page are a little different, here you can find a map of widget locations in the free theme: MH Corporate basic.
I know a little bit of coding and was trying to figure out where in the theme it generates the order of which section (author block, nav) etc gets created. I found the actual creation sections (mh_post_content), but I’m not sure what chooses the order.
Your plugin is using a content filter to place the social buttons right after the content and the author box is placed by using a hook. You would either need to use a filter to add the author box as well or disable the plugin filter and hook the social icons after the author box. In either way it would be recommended to create a child theme.
Here you can find more information about hooks and filters: Action Reference – Filter Reference.
But please keep in mind that you are using a free theme without any special support packages included and we can’t offer free support for theme customizations.
Have a nice weekend.