When Goat dies, after the last attempt at defibrillation is made and the others give up, you can see the supposedly "dead" Goat take a big breath just before Reaper says "He's gone". In addition, in this shot Goat's eyes are wide open when the camera is on him. When the shot goes from Reaper back to Goat, his eyes are now closed.
The B.F.G. is a weapon that dates back to the game "Doom" back in 1993. There, it was the 'ultimate' weapon which, when fired, cleared the screen of foes in a great sickly green blast, consuming enormous amounts of ammo. It has showed up in various later games by both ID Software and third parties. Up to this movie, the acronym was never explained directly - although Sarge's version, "Big f**king gun" was for a long time the community's accepted name (also used in the original Doom design document, but not the game itself), and is most likely a homage to that.