The point of triage in an ER is to identify and treat patients in need of immediate care. No nurse in his/her right mind would pass over a girl having an acute asthma attack for an obviously stable man with a gunshot wound, no matter what order they arrived in.
When the homeless guy is speaking to Arlene in the garage you can see that his teeth are good. During the shot when a woman wants to jump from the bridge, this guy goes to her and starts smiling, it's totally seen that his teeth are bad and dirty.
In the beginning Eugene writes the task on the blackboard, and later in the movie he is cleaning it from other notes and clearly the task given is not on the blackboard. Near the end when doing the interview you can see the task once again written on the blackboard, as if it had never been removed.
After Arlene slaps Trevor, she goes into the garage looking for stashed alcohol. On the right side of the shelves she searches, there are two boxes of powder laundry detergent stacked one upon the other. Although each would weigh a few to several pounds even if only half full, she brushes them off the shelf with just a flick of her hand, revealing the boxes to be empty.