I found the below answer from Danny most helpful for putting social icons in menu bar.
Now I would like to put a mail icon in the menu.
Please tell me how to do that.
Thank you Danny. I never would have figured that out. See below.
Unfortunately, there is no option to include both the menu and social icons via the Customizer, unless you select the Nav Bottom option.
However, what you can do is navigate to your WordPress Admin Dashboard > Appearance > Menus and select the menu that is being used in your header.
From here, click on the Custom Links module and in the URL field, add the link to your social network i.e. your Facebook page and in the Link Text add <i class=“fa fa-facebook”></i>.
Repeat this process for all you social networks such as:
Twitter = <i class=“fa fa-twitter”></i>
Google+ = <i class=“fa fa-google-plus”></i>
For a full list of Font Awesome social icons, see the Font Awesome website.
When you have completed adding your menu items for each social network. Save the menu and when you visit your sites front-end. You should have something like this: