Menus are created in the Menu Editor at Appearance > Menus, then assigned to one or more menu locations, either in the Menu Editor or in the Customiser in Menus > Menu Locations.
The theme has two menu areas, Header Menu and Footer Menu. The Footer Menu has no drop down option so is restricted to first tier menu items only.
The settings for styling of the Main Menu are in Theme Settings > Main Menu.
By default, the menu bar area includes the site name, with a Customiser option to replace this with a logo in the Site Identity section.
With either of these options, the menu can be moved to the right. This is done in the Customiser under Theme Settings > Main Menu > Menu Position.
Another option is to replace the Title / Logo with a widget area. This can be enabled in the Customiser under Theme Settings > Main Menu > Main Menu Widget Area.
The widget area, Main Menu Widget Area can then have a widget added in Appearance > Widgets or in the Customiser in Widgets > Main Menu Widget Area.
The widget area is always right aligned, so will over-ride the Menu Position setting.
There are two recommended widgets to place in this widget area: Simple Social Icons or the standard Search widget – other widgets may work but might need code customisation.
The recommended settings for Simple Social Icons widget are an icon button size of 36px and border radius of 3px.
For a search box, just place the standard WordPress search widget in the Main Menu Widget Area.