a) Memory Alpha lists Mike Okuda as the person making Chapel a brevet ensign, in a game that came out 1998.
b) There have been nursing schools in the United States since the late 1800s . By the 1930s the Yale School of Nursing was offering Masters Degrees in Nursing. I imagine most of what the folks in charge of Trek thought of Nurses came from encountering them while in military service in WWII.
Chapel is mentioned to be a student of Korby's in WALGMO. Korby's last message was five years before the events of the episode and Chapel signed up for service aboard a starship hoping find him. So I'd guess Chapels been out there looking for him for those five years. What it doesn't say is if Chapel was in Starfleet prior to Korby's last message. Could be she was in the Science Division as bio researcher. I hope she didn't Kim it and stay an Ensign for years.
c) Where does it say she was promoted to Lieutenant? Pretty sure she just shows up with a Lieutenants stripe. Lt jg is a dashed stripe, so she was a full Lieutenant in TAS.
d) Noel is a Doctor, at best she'd have to be a Lt jg like Bashir and sport the dashes. Anne Mulhall was a Lt. Commander working as an Astrobiologist. She could have answered to McCoy (Lt. Commander) or Spock (Commander). As a redshirt she might have answered to Scotty (Lt. Commander) though lord knows why.