If the extractor fan is causing smoke to come back into the room then there is a big problem with the install. A vent should have also been put into the room where the stove is.
An extractor fan should not have any bearing on the operation of a flue/chimney. A cold draught coming down a chimney is remedied by building the fire slowly to warm the chimney and get the draw going. Has there been a new pot fitted? A gas terminal on a pot with solid fuel fire below is NOT...