It just looks like a custom menu with a fancy effect on it.
If you dont need the fancy effect and the tab look you can get much the same arrangement of drop down menus by just making a custom menu in APPEARANCE > MENUS and then creating a menu linking to all the pages/categories you need.
If you actually want the fancy styling I’d suggest you hire a developer though to make css customisations for you.
Are you sure its the theme?
Widget functionality should not be affected by anything the theme.
Temporarily switch to a different theme to check.
I’ve not tried the plugins you mention lately but I’ve found the widget per page functionality now built into Jetpack plugin works well.
