The sequel has a quick joke about the "spread of feline extremism." This helps to explain why cats were originally regarded as evil- It's the equivalent to how so many Americans think of Arabs. And of course, even if the majority of enemies we face all are the same race/religion, they are only the worst of the bunch. So Catherine's like an Arab in counter-terrorism.
A quick comment by Catherine indicates that MEOWS' primary purpose is to protect humanity from canine threats (their claim of jurisdiction on Kitty Galore is her being ex-MEOWS), just as DOG focuses on feline threats. Of course both sides are going to be reflexively suspicious of the other: they keep on meeting the worst of the other species.
Why does every installment have a different cast? Being ordinary animals, they have pretty short lifespans.
Where do you think Kitty Galore got her puppy outfit from?
Take a look at what Kitty Galore's evil plan was. Realize that the key device wasn't shut off until the effect had traversed most of the western hemisphere...
Also counts as Genius Bonus, At the end of the Ancient Egypt segment, it shows Shen-Akh-Akumon and his Mooks being fed to crocodiles. In Egyptian Mythology, souls of the damned were eaten by Ammut/Ammit, who was a female demon made of Egypt's most dangerous animals, including the head of a crocodile, the midsection of a lion, and the rear-end of a hippopotamus. She would lie under the scales of truth to see if the soul was evil. With this being said, it's possible the dogs themselves killed Shen-Akh-Akumon and his followers, and given the type of pharaoh he was, they were eaten by Ammit in the afterlife.
Ivy doesn't appear in the second movie, possibly meaning she died between the events of the first and second film.