Can I show an icon next to a menu link?

Yes. On Settings → Appearance → Layout Settings, Menu tab:

1. Expand the menu item under Menu Structure.
2. Set Icon type to Font Awesome or Image.
3. For Font Awesome, type the icon's class in Icon, for example fa fa-fire. For Image, click Choose and pick a small picture from your media library. Use remove to clear an image.
4. Set Placement to Before label or After label.
5. Click Save Menu.

When you add a new item under Custom Link, the same options appear, with an icon list to pick from and an Icon placement setting.

Keep image icons small and square so they line up with the text.