How do I make sure every glyph has the same width?

Since 3.1.1, you could use the icon-fixed-width class instead of having to edit the CSS.

http://fortawesome.github.io/Font-Awesome/3.2.1/examples/#navigation

Since 4.0, you should use fa-fw:
4.x https://fontawesome.com/v4.7.0/examples/#fixed-width
5.x https://fontawesome.com/how-to-use/on-the-web/styling/fixed-width-icons

Thanks @kalessin for pointing out.