The comic relief subplot between Dinky, Boomer, and Squeaks mirrors the larger plot in several ways.
The way Squeaks hides in Widow Tweed's house all winter reflects how Tod finds safety in the game preserve for a while. Similarly, the part where Dinky and Boomer try to trap Squeaks by blocking off both sides of the drainpipe could foreshadow how Amos and Copper try to trap Tod in his den the same way.
At the end, Amos spares Tod's life, Chief presumably no longer hates Tod, and his friendship with Copper is restored. When Squeaks turns into a butterfly and Dinky and Boomer decide to let him go, it reflects this reconciliation between hunters and prey.
When Copper starts leaving his barrel to go play with Tod, Chief makes a point to tell him how upset Amos Slade 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 Slade treated Chief and Copper? 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 sight hound. Sighthounds are bred for just that—flushing and chasing prey based on line-of-sight, working in tandem with the handler. Scent hounds 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.
Another bit of Fridge Brilliance is that those types of dogs are often used together.
Slade is completely taken aback when Tod and Vixey jump through the burning grass blocking their escape from the foxhole. Any wild animal would instinctively flee fire, but Tod was semi-domesticated, and had been by Widow Tweed's fireplace before.
Naturally, Copper is frightened when he gets the scent of the bear. Bears are The Dreaded and even game hunters consider them dangerous, so he was no doubt trained to avoid bears.
The movie 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.
It's a good thing this is a Disney movie, or the movie would have ended as soon as Big Mama saw a newborn fox cub left by its mother. Great horned owls eat fox cubs if given the chance in real life.
While trying to get Tod to understand the harsh reality that Copper, as a hunting dog, will have to become his enemy, Big Mama, Dinky, and Boomer show him Amos's shack of animal skins. At least one fox pelt can be seen.
Anybody wanna bet that there's a possibility that that fox pelt could've been the remains of Tod's mother?
Unlikely. Mere minutes after Tod’s mother dies, we see that Chief was asleep and Slade is only just getting home with Copper. It’s highly unlikely that he stopped to hunt a random fox, without his hunting dog, while he’s got a puppy in the back of his truck.
Between the burning grass and Copper, Tod chose to try his luck by jumping through the flames. Copper used to be Tod's best friend, and now he became scarier than the fire. Then again, facing Copper also meant facing Amos's shotgun. Tod took what he considered the safer route through the fire (see Fridge Brilliance above).
When the bear hits Tod or Copper, he sends them flying, but when he hit Amos all he did was knock him down the hill. Amos is heavier than them, but bear also uprooted a dead tree, so that probably didn't factor in. But given how lightly he seemed to hit Amos, it gives the hint that the bear was just trying to knock his gun out of his hand. Between that and how the bear slows down when it looks like his victims are down for the count and winds up more before trying to strike them (and rather oddly only uses his teeth once). He paints the bear as especially cruel and that he enjoys frightening his victims before killing them. After Amos shot him, the bear wanted him to feel helpless in retaliation for shooting him.
Most material would have you believe that the trap Amos stepped into was one of his but in fact it wasn't. It was a trap from a previous hunt and might even have been for the bear itself. Which brings up the point of what happened to that hunter—the bear seems to instinctively go into kill mode at the mere sight of Amos and is determined to end him even after he's no longer a threat where most bears will break off the pursuit at this point.
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.
. ("A spider'll not even notice a drop like this, a mouse'd walk away, a horse'd break every bone in its body and an elephant would spla-"). In other words, the bigger they are, the harder they fall.