Duke asks Rosa for her telephone number when he takes her home. But he has already telephoned her home to make the date.
When Detective Ferguson is reading Duke Martin's criminal record, it shows Duke's Date of Arrest as 7/2/44, but his Date of Birth as 11/14/44, indicating he was arrested 4 months before he was born.
The scene where Duke asks for Rosa's telephone number (which he already knows) is simply a device to remind non-observant members of the viewing audience of that fact.
When Detective Ferguson is first questioning the two beauticians, a lot of emphasis was placed on the second gunman's hair colour. Part of the beauticians' business to know hair colouring (including eye brows). The plot focused heavily on the differing statements from the two. The police didn't reference or question the height of the second gunman, who was much taller than the person they were trying to frame.
When the witness calls Rosa's telephone number to speak to the policeman, Duke overhears and thinks Rosa has betrayed him. In fact, the policeman has given the witness the number. It is necessary to the plot for Duke to think that Rosa betrayed him, but it is absurd that he would give her the phone number, since he would be there for only a few minutes during the day, if at all.
Reading from a book, Jackland Ainsworth quotes, "Some women should be struck regularly - like gongs", adding, "That's from Oscar Wilde, you know." In fact, it's a quotation from Noel Coward's play, "Private Lives".
The police sergeant is startled when Rosa identifies herself, as he hasn't seen her since she was very young and her appearance has greatly changed. But he is friendly with her mother and brother, who live with her, and appears to have seen them recently.
Rosie giving out her phone number to Duke when he dropped her off home the first evening. He didn't know her number previously (the number becoming significant later). The date was arranged after he dropped a note off with his invitation written on it.