At the end, when the elusive caterpillar that Dinky and Boomer chase throughout the film turns into a butterfly, it's a reflection on the larger plot where the hunters and prey reconcile.
When Copper starts leaving his barrel to go play with Todd, Chief makes a point to tell him how upset Amos will be about it. True to this prediction, Copper is put on a leash for his misbehavior. Before Copper started wandering off, what was the biggest difference in how Amos treated Chief and himself? Chief was always kept on a leash. He had been trying to warn Copper about losing his privileges the same way that he did when he was younger.
Chief is dramatically outshone in tracking abilities once Copper shows up. Old age? Not likely. Chief looks like a shaggy greyhound-esque breed or mix—lean, tall, long thin muzzle, small ears, no wrinkles. In other words, he's a sighthound. Sighthounds are bred for just that—flushing and chasing prey based on line-of-sight, working in tandem with the handler. Scenthounds like Copper have been designed through thousands of years of breeding to locate prey by smell alone. They're just built for different styles of hunting. Copper was always going to be better at the kind of tracking-heavy hunting style his master prefers.
Although Tod was Demoted to Extra in the second film and Copper was shown to shine in the film, if you think about who was the protagonist and the most important character in the first film, you'll realize that the focus merely switched over from Tod to Copper.
Why did Amos keep Chief on a leash but let Copper go unrestrained? It's all well and good to teach your dog to stay near your house on its own accord while it's still young, but shouldn't Amos have been personally watching him during this learning period? There are wild animals all around their house! A puppy Copper's size could geteatenif it wanders too far.
Is Slate going to be arrested for trespassing onto a game preserve? Hunting license or not Slate was poaching because of a petty vendetta.
When Tod is getting to know Vixey, Dinky and Boomer suddenly show up out of nowhere to watch them.
During the bear attack, Amos struggles to reach for his gun since his foot is caught in one of his traps. The trap is clearly seen being held down by a post; it would probably take less effort to pull it out of the ground and go for the gun rather than try to pry the trap open.
The Fox and the Hound begins with the baby fox's mother hiding him and sacrificing her own life for his, Bambi-style. However, foxes almost never have a single cub. This mother wasn't just carrying her baby to safety. She was carrying to safety the only baby of the litter who survived the hunt.
Although it's more like Fridge Sadness: The end of the film pans out on Tod watching Copper from a hill. It was earlier shown that the widow drove what was possibly hours getting Tod to the reservation, meaning he traveled a long while to look at Copper and his old home one last time.