backfires when running and let foot off gas

Typically, if it's backfiring out the exhaust on deceleration, it's because of a diverter valve for the air injection that is stuck or not receiving the proper vacuum signal. It could be dumping fresh air into the exhaust when it shouldn't be. The Idle: some possible causes are an idle set too high causing the EGR to open and close and perhaps float settings. It sounds as though it needs a little TLC. Often, the exhaust manifold warps causing the intake to warp with it and creates a vacuum leak along with the exhaust leak. Both manifolds are bolted together and are quite heavy to hang off the side of a motor without checking the torque of the nuts a couple of times a year. Someone sets the idle high to try to keep it running with a vacuum leak and the EGR activates. Kind of a cascade effect.