Visible crew/equipment: When Punjab is remarkably "flying" the toy plane by using his mind and interesting hand movements, when it shows a close up of the floating plane, you can see the string to which it is attached.
Visible crew/equipment: In the rescue scene with Annie and Punjab on the bridge, when Annie is climbing the ladder, in the wide shot you can see it's not a girl but the stunt double, quite possibly a small man.
Answer: Mrs. Hannigan's box of Annie's parents' things might have the only record of her birth certificate. This is the 1930s, after all. Paper records went missing all the time.
Captain Defenestrator