I remember trying that... linger a little while packing up in hopes she will say something as she walks by on the way out. In hindsight that is weak, either wait for her and speak to her directly, or just leave the room.
I don't think he really cared about apologising, just wanted an excuse to try to stirr up conversation, I have done similar.
Don't get too excited about a friend request, in my limited FB experience EVERYONE makes EVERYONE A friend, more the merrier.
As I see it , the ONLY thing you can do is ask her out on a date.
To the rest of the forum, would it be acceptable for him to message her back something like... "that sounds great [chatting], how about over dinner, i'd like to take you out" ? I know that is pussified over the Internets, but may be good considering the level of lingering (and shyness, been there too) we have here.