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' 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.
There's also the fact that the dog crisis was manufactured and contained within Megasaki. Other parts of Japan don't seem to have the same issue with dogs.
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.
Tracy's declared hatred of Mayor Kobayashi might seem a bit narmy and over the top. However, her behaviour makes more sense when it is revealed that her dog (Nutmeg) is also on Trash Island. Indeed, Mayor Kobayashi is very lucky that his opposition has stuck mainly to verbal arguments and nobody has gone JohnWick on him.
The wasabi poison being so potent that Professor Watanabe literally barely disturbing the minuscule smear of a minuscule smear with the very tip of his tongue dies with more poison in his body to kill a goddamn whale seems like overkill of the highest order, but Kobayashi most likely knew that the Professor would be smart enough to notice if there was any poison in his sushi but curious enough to taste-test it, which is exactly what happens.
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).
Actually, most of Duke's rumors are eerily accurate, although this is more likely due to dumb luck than genuine knowledge on his part. The Aboriginal Dogs ARE cannibals (even though they aren't happy about it) Atari is exactly who he claims him to be, King's dog food chain did indeed shut down, and the Dog flu is implied to come from the facility where the lab dogs were tested. As for Nutmeg and Felix, we only have her word on that...
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' 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.
Actually, their names (or some of them) being in English might be the said English equivalent (or closest thereof) for their Japanese names. Writing it in a Western alphabet is just for convenience.
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?
The dogs have all been noted as being translated from "barks to English." It could be that, in-universe, there's some canine term for it that doesn't translate especially well to English. That, or "sister-in-law" was just the easiest way to put it as opposed to something longer and more convoluted like "the mate of my youngest littermate," or it's a bit of a more distant relation like "the mate of the middle brother from a litter twice removed from mine."
At the end of the film, Mayor Kobayashi, who has had Heel–Face Turn, is sent to prison with his co-conspirators, whom he betrayed, ratted-out and all still unrepentant. What are they going to do to him in there?
The dogs were on Trash Island for six months. Many dogs died trying to survive there, meaning that their owners who may have still loved them never got to see their pets again. The opposite are also true, as some dogs who may have survived the island may have had owners who passed away during their time away.
