Bathroom vent stains are from moisture condensing in the vent-pipe when the attic and pipe are cold enough. Hopefully the air exhausts outside the house, not just dumping warm, moist air in the attic. The vent hood should have a flapper to reduce the flow of cold air back into the house.
To stop the condensation, insulate the pipe or replace with insulated venting. It is a three- or four-inch pipe and it could be either metal or plastic. Wrap the vent pipe with fiberglass pipe wrap. It comes in rolls of 25 feet with a plastic vapor wrap and depending on the distance you could require 3 or 4 rolls to complete the job. Overlap the seams as you wrap the pipe and do not stretch the material too tightly. You get more insulation value when the fiberglass is still fluffy.
While you are in the attic you should caulk around the exhaust fan and the ceiling. This will help prevent warm moist air from leaking up into the attic, possibly condensing on the drywall or plaster and causing paint damage.
Paint peeling from a ceiling is often caused by poor ventilation and moisture in the attic. Also, check for poorly fitted light fixtures. A stream of warm moist air into the attic could be dampening an area causing paint to peel.