Your ceiling will need proper preparation for the best adhesion of the paint and this does include removing all the loose chipping paint.
The first step is preparing the area for repair and painting of the ceiling. Remove as much furniture as possible and cover any remaining with light weight plastic. Cover the floor with drop cloths. The age of your home guarantees that some of the old paint layers have lead. For this reason use a good lead certified particle mask, eye protection and wear either old clothing or a disposable painting suit, AKA "bunny suit".
The loose paint and plaster needs to be scraped off, use a stiff 3 inch putty knife or five way paint scraper. Once the loose paint is removed, saturate the area(s) with bleach and water to kill the mold. Allow the ceiling to dry for a couple of hours then prime the affected areas with a good fast drying acrylic or oil based primer. The primer will stabilize the plaster for the next step.
Patching the area(s) is next. Use drywall mud, either regular all purpose or setting type (powder). The setting type of drywall mud is what I like to use, but is only available in large bags. Apply the patching compound and sand it until the area is level and smooth with the surrounding ceiling. Once it has fully dried, prime the entire ceiling to seal the patch and provide a suitable surface for painting.
Now the ceiling is ready for painting. It seems like a lot of work and it can be depending of the condition of your ceiling.