“Michael Gennaro, who announced last week that he would be leaving his positions as president and chief executive of the Paper Mill Playhouse in Millburn, N.J., will become the executive director of Trinity Repertory Company in Providence, R.I. He is to succeed Edgar Dobie, who is leaving in January to focus on his duties as executive producer of the new musical ‘The Pirate Queen.’ ” Gennaro’s move to Trinity means a reunion with his former Steppenwolf Theater Company colleague, Curt Columbus, now Trinity’s artistic director.