Re: Why women dosen't display their Ranks ?
Pardon, but I don't remember any dialogue saying anything about her rank, just her medical credentials.
I don't recall any such dialogue, either, but a professional degree typically goes with a certain minimum rank. Many Trek characters with a doctoral title carried the rank of Lt. Commander, though fresh-out-of-medical-school Dr. Julian Bashir was a Lieutenant, so I'll amend my above statement to say that, at the minimum, Dr. Noel should have been wearing a sleeve stripe for that rank.
"It is my belief that nearly any invented quotation, played with
confidence, stands a good chance to deceive." — Mark Twain