Continuity mistake: When Joe is hiding from the Gat behind the couch, his gun swaps from facing a 3 o'clock direction to a 1 o'clock between shots.
Continuity mistake: When the waitress delivers the identical breakfast plates to the two Joes, she places them in exactly the same way with respect to the men - eggs toward them, steak facing away. In the longer shot, old Joe's is different, although neither has touched the plate.
Continuity mistake: In the scene where Old Joe (Bruce Willis) is forced to the time machine and then fights back, the amount of Gat Men changes. 2 are seen bringing him in, another collecting the gold and the brown jacket, and a fourth opening the time machine. However, when he fights back there are only 3 bodies. It is plausible but unlikely that the fourth man would have left and gotten far enough away not to hear the fight.
Continuity mistake: When Cid is in the field, his face is sometimes covered in blood, or with a clean trail on the right cheek. This changes randomly.
Continuity mistake: When Cid is comforted by Joe and his mom arrives. Cid is either facing her or the opposite side, depending on the shot.
Continuity mistake: While hiding from the Gat man in the tunnel made by Cid's grandfather, the stick used to hold open the trap door switches from the knot being at the side, to the knot being at the front.
Chosen answer: The first time the time-loop occurred it is not necessarily due to Bruce Willis' character killing her. She could have died in another way, instigating the timeline in which Bruce Willis is taken to close his loop. He knows who the rainmaker is and therefore attempts to kill him in order to preserve his happy future. This in turn creates a brand new timeline in which Joe kills himself to save Sara, who in turn prevents the rainmaker from becoming a crime lord.