I'm wondering if any closers have had a game winning hit or home run? Since this requires the closer (pitcher) to be at bat, it implies that the game was hosted by an NL team.
Yes, there actually has been a pitcher to hit a walk-off homerun. This took place on April 25, 1986. Craig Lefferts (a left-handed pitcher and batter) hit a walk off homerun in the 12th inning while he was on the San Diego Padres.
Jim Hardin also hit a walk off home run as a pitcher on May 10, 1969. At the time he was playing for the Orioles and he hit it off of the Royals