Okay, walk me through this - do you, say, see a cute girl (friend of a friend, someone in your city, etc.) that you might fancy, and just randomly send her a message?

Yes. You casually introduce yourself, mention a thing or two in common (same groups for instance), and ask if they'd be interested in getting to know each other —assuming you have a solid FB profile and not something fake. Pretty much like you'd walk up to a person in a real life in a social setting and say Hi. Many of my friends met their girlfriends like that. If you don't get a reply then that's your answer, no hard feelings. If they're a friend of a friend, well, you just ask your friend for an introduction. I don't know how things are in the US, but it's not considered awkward at all in my country. I've had women approach me in this same way as well.

This is mostly between friends-of-friends , not for randomly adding people for hookups. Typically it involves sending a friend request followed by chat