J.J. Abrams’ Bad Bad Robot Productions will stay with Paramount Pictures through 2013. Paramount recently locked the deal as an aggressive promotional push accelerates on “Star Trek.” Abrams directed and produced the film, which opens May 8. The studio first partnered with Abrams and producer Bryan Burk in July 2006. Their first venture was “Cloverfield,” grossing $170 million worldwide and leaving a doubtless sequel in its wake. “Star Trek” marks their second effort, followed by the Harrison Ford picture “Morning Glory,” which begins production this summer for a 2010 release. “Bad Robot was one of the first deals we made to help rebuild the studio,” Brad Grey said, Paramount Pictures chairman and chief executive officer. “J.J. has since become part of the fabric of Paramount, and …