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.

For Fontawesome version above 4.X, use fa-fw class in the <i> tag. Ref:- https://fontawesome.com/how-to-use/on-the-web/styling/fixed-width-icons

Its simple and easy to scale glyph or any icon using this css

> .fa { transform: scale(1.5,1); }

Are you sure that you haven't got another style defined that is doing this?

This is how your HTML looks placed into a file on a site I have using Font Awesome:

Notice how the icons and the text line up. This is your original image with lines added:

It looks like you have a style defined somewhere that is removing the Font Awesome styling.

You could also try adding in the original Font Awesome style (coming from font-awesome.css) to see if that solves it temporarily:

li [class^="icon-"], .nav li [class^="icon-"], 
li [class*=" icon-"], .nav li [class*=" icon-"] {
    display: inline-block;
    width: 1.25em;
    text-align: center;
}