The dogs speaking English, despite residing in Japan, could be interpreted as the language barrier between dogs and humans (who themselves speak Japanese). For dogs, not understanding human-speak is like being an American who doesn't speak Japanese. This is explicitly said at the start of the movie.
At first, it seems somewhat pointless that there's a plot twist where Spots and Chief are long-lost brothers. But when you think about it, in a way, this genetically makes Chief Spots's worthy successor as Atari's guard dog.
However, there may not be as dire situations in other countries, which are not mentioned to have any problems with dogs. It's possible companies from overseas will be able to enter the Japanese market, thus accommodating the dogs.
Additionally, other Japanese dog food companies may have changed their plans to export their foods to other countries to prevent going out of business.
The mayor's guarding of Atari with home schooling, a guard dog, and keeping him mainly in the estate makes sense given all of the shady and illegal things the mayor was doing. Eventually, someone could come after Atari, and the mayor wanted to minimize the boy's harm.
Chief's hardened but kind / protective personality and his skills at fighting comes from being bred to be a professional guard dog.
A lot of the rumors Duke picks up are sensible to be circulating among the stray dogs (Nutmeg supposedly mating with Felix, cannibal dogs roaming about, a dog committing suicide, etc.) and the one about Doggy Chop folding is a logical conclusion, what with all the dogs gone from Megasaki. But what about the rumor of Atari being the mayor's relative and ward? There's no reason for that nugget of information to be floating around, right? Well, later in the film we see a flashback of Spots talking to another dog, describing his duties as the mayor's nephew's bodyguard. So when Rex began asking around about a dog named Spots, whose owner had come looking for him, word got around as the dogs of Trash Island put two and two together, which Duke would later hear.
To some extent, but this is anti-Fridge Horror: Duke claims his brother committed suicide by hanging himself with his own collar. However, the act would require hands, something dogs unquestionably lack. This can be speculated as untrue, especially when you take into account the Running Gag that a majority of the 'rumors' Duke hears often prove untrue (like Nutmeg mating with Felix, or the Aboriginal dogs being cannibals).
Also note how the dogs seem to have a pretty good grasp of how to operate simple mechanisms, and Duke specifically says his brother killed himself with his leash. If he had his leash still attached to his collar or was able to attach it himself, he'd simply have to find something slender and high off the ground or overhanging a drop to loop the handle through...
We never get an explanation as to how some other dog's bones wind up in what Spots's cage. The lock on the cage may or may not be commonly available (the dogs in the former research facility had a key, and Atari's key worked on the dead dog's cage), but we're merely left to speculate on this. Especially since we don't hear about any other dogs dying in such cages.
Atari's key may be a skeleton key of sorts, as the cages would have all been produced by Kobayashi. It's likely it is simply a different cage that happened to land in the same, or a similar, place to where Spots landed, especially as Spots was transported in his cage to the other end of the island, and it wouldn't make sense to take his empty cage all the way back to where he landed.
Likely just coincidence. Two dogs, one being Spots and the other being the dead Sport, just happened to wind up in the same type of cage. Sport was just unlucky enough to have nobody find him and rescue him.
Putting aside Translation Convention, why do all these Japanese dogs have English names, written in Western alphabet no less?
It's not altogether too uncommon to name your pet something from a different language. And there's a decent popularity of American culture in Japan. It's not unthinkable that they could have English names. Especially since at least two of them were mascots for a company and a school sports team respectively. As for why all five of the dogs would have those sorts of names... well, consider it just a thing of the movie.
How did Atari understand the dogs near the final act? Earlier on the film it's easy to imply that that dogs are smart enough to recognize his commands, but towards the final act you get things like how he manages to understand Spots and Chief trading places or how he knows of the wasabi poison plan delivered by the owl.
Nutmeg refers to her sister-in-law. So dogs can get married?