Jaime addresses Roger Grette by his first name when they first meet at the ranger station door, so she would not address him as "Mr. Grette" later, especially since they obviously were now fairly comfy with each other, and thus would probably expect to use each other's first names.
The "dog may be dangerous; shoot to kill" note that Roger had jotted down sometime earlier from having heard the fugitives report on the ranger station radio neglects to include the extremely important provisional "if the dog attacks". An experienced law-enforcement officer like Roger would *always* be sure to take down all the vital details, especially since he was writing the note not just as a reminder to himself, but to his fellow ranger buddies, too (just in case they had not also heard the APB broadcast on their portable walkie-talkies), and so they would need to know not to just automatically shoot Max if they merely saw him, but only if he actually lunged at anyone.
Max rips the door off its hinges as he bounds outside after being frightened by the cooking-alcohol flames, yet as Roger and Jaime head out to look for him, the door is neatly attached to the frame again. This scene takes place just a relatively few minutes after Max escaped, so Roger and Jaime would not have had a chance to re-attach the door, nor would they have wasted valuable time during their phoning Oscar and getting ready to head out to look for Max to attempt any reparative carpentry.
A police cruiser and a late-60's-era white pickup truck are shown parked in front of the gas station as the truck that Jaime is riding in is heard pulling up, yet when the truck slows to a stop at the pumps, both other vehicles are gone. Then when the truck driver is talking with Jaime, the police car is back, but the white truck is still gone.
Max runs several yards down the track till he comes to the flaming blockage, so he would be too far out of earshot --- especially with the loud roaring and cracking of the forest fire all around him, which would drown out any faint sounds, even if Max had bionic hearing, which he does not --- to be able to hear Jaime's weak/hoarse orders to go and bring the rescuers.
While communicating over the radio with Fire Boss, the helicopter drops its load of foam twice in the space of a few seconds. Firefighting helicopters can only carry a relatively small amount of extinguishant at a time, so this chopper could not have dropped two large doses in one flyover, as it is shown doing here.
One of the firefighters radios to Roger to inform him that the fire has now ringed the top of the mountain, yet the map on the wall of the ranger station already has a wide red circle drawn on it. This was the first time that Roger or any of his fellow rangers in the room had heard about the fire's surrounding the top of the mountain, so nobody there could have known to draw the circle on the map yet.
Shortly before Jaime yanks out the shorted wires on Max's neck, a close-up shot of the damaged circuits sparking reveals that the surrounding area of "fur" catches fire, whereas it is not shown burning on the dog's neck in the wide-angle shot immediately following.
After the wolf kills one of the sheep and Max comes over to fight off the wolf, we see the other sheep bleating and wandering away in fright. But then several larger sheep are visible on the left, calmly standing still and seen in sharper focus than the retreating animals. So obviously this is just a "rear projection screen" shot, which shows a small group of live sheep on the left who are watching a movie of some running sheep that is being projected on a screen in front of and to the left of them.
Max places his paw against the lower jaw of the trap to hold it down so that he can bionically pull up the top jaw with his teeth to release the man's foot, but when he starts to pull the top jaw up, he removes his paw from the bottom jaw and it stays down; obviously this is a fake trap without a spring.
The forest fire is shown to have blackened and consumed the trees that the train is passing through, yet when the firefighters and Oscar and Rudy arrive at the stopped train, there are green-needled branches all around the area, with very little of the trees burnt or even actually on fire.