I don't remember whether I sent a particular person on Facebook a friend request. The option to add them as a friend is open which either means I did and they rejected it, or I didn't. I don't want to send them another one If I did already. They still show up in my "People you may know" list. Would they still show up if I they rejected my friend request?
No, they will not show up in your People you may know list. Because when a person rejects your friend request, you will not have the option to add them as friend again. So, Facebook doesn't show their profiles in your People you may know list.
But they will have the option to add you as a friend, so probably your profile will show up in their People you may know list.