The Annabell thing wasn't random.
Right after you start getting attacked, you find a voxophone from that same woman who's looking for Elizabeth. I assume she was just asking every girl she knew until she got a specific name, since she wouldn't be able to immediately confirm it was Elizabeth otherwise, not having seen her before.
Sure, it required some breakdown, but it definitely wasn't a random thing.
What? They time travelled CONSTANTLY.
And all the events leading until old Elizabeth's interference was by the whims of the Lutece's. It was only after they "died" that Booker was brought trough time and space to Columbia.