During the elevator scene, Cain boards on the 6th floor, and the elevator travels quickly to the 5th floor, where the other man boards. The entire scene with the new passenger, including the fight, takes place after the door closes on the 5th floor, but before it gets to the 4th floor. It then descends to the 1st floor in the space of seconds.
1 hr 9 minutes into the film when Cain and Anna tried to leave the hospital room, two shooters machine gun the door full of holes stopping them. When they left the second time they shot the shooters and the doors do not have any bullet holes.
The scene that takes place in an "Arizona DEA office" the flag behind Nick Nolte's character is for the state of California.
20-minutes in, Cain is firing a rifle in a hallway. The camera is behind him and over his right shoulder as he fires. It switches to slo-mo and you can clearly see the "spent" rounds being expended from the chamber are intact, rubber-tipped prop rounds meant to mimic the expended casings.
Alternate Explanation:
It appears that the shell casings the AR-15/M-16 series rifle is ejecting are actually from blank rounds. Instead of a rubber tip, the blank round has gunpowder in it, and they crimp the tip of the casing. This allows the rifle to fire, eject the round, and with the gas from the round, cause the next round to load, just as if it were a live round. This also explains why there is no muzzle flash since, for the process to work, they must plug the barrel.