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: Miss 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