Ratigan surrounding himself in pink wallpaper and having pink as part of his wardrobe seems at first glance to merely be Disney making him look like a Sissy Villain; until you remember that in 1897, pink was considered a more appropriate color for men than for women; ergo, in his eyes, a pink hideout is a masculine hideout.
Early on in the film, we have a funny moment when Basil tries to get Toby to sit, but Toby doesn't listen until Olivia orders him to sit. It's funny but also makes a lot sense if you're familiar with Basset Hounds. They are notoriously stubborn and respond well to food and other positive reinforcements. What did Olivia do upon their first meeting? Feed Toby a crumpet and rub his tummy. No wonder he responded well to her command to sit.
Fridge-Awesome: It's cool enough that Basil and Ratigan's final battle takes place atop Big Ben. But the climax of their fight becomes even more meaningful when their fall to their deaths is a homage to "The Final Problem" (the short story where Holmes and Moriarty fight to the death at Reichenbach Falls).
When Basil rightly suspects that his drink is spiked, he sniffs it and tastes a tiny bit. This is the exact behaviour of real-life mice and rats when they suspect poison; they're "bait-shy". (This is why rat poison is so dangerous - it has to be lethal even in small concentrations to kill from a single taste)
Oddly enough, there is a bit of Actor Allusion mixed with Fridge Brilliance in regards to Ratigan's death-trap Rube-Goldberg device: One of Vincent Price's other roles was fond of doing such traps as The Abominable Dr. Phibes. They even have a phonograph projecting the character's voice as well.
The lady at the end says she's missing an emerald ring? It's on the finger of the Rat Trap bar pianist.
Basil has so far been unable to collar Ratigan until meeting Olivia and Dr. Dawson. Paying close attention to the events of the film reveals that while Basil is brilliant at deduction when he has information to work with, he often misses obvious clues in his search for details that Dawson picks up. This is almost exactly how Holmes and Watson work. Dr. Watson fills in the gaps of Sherlock's practical knowledge since Holmes doesn't bother to learn anything not related to criminal science.
Believe it or not, Dawson's What the Hell, Hero? speech to Basil as the latter is suffering from a bad case of Heroic BSoD and doesn't actively look for a way out of their deathtrap also is similar to Holmes and Watson's relationship. Thanks to Nigel Bruce's portrayal of Watson against Basil Rathbone's Holmes, many people have the mistaken notion that Watson in the Holmes Canon hero-worships Holmes and sees him as an absolute paragon of virtue...when in reality, there are a few times when Watson chews Holmes out for his self-destructive tendencies (mainly involving his use of morphine and cocaine as recreational drugs to cure his boredom between cases) and even argues with Holmes about refusing to learn common sense things in A Study In Scarlet. And, like Basil does in this scene, Holmes listens to Watson. It doesn't mean he always takes his advice to heart, but Holmes respects Watson enough to at least heart him out.
One of the phrases Ratigan says to Basil when he leaves the depressed detective and Dawson tied into his Death Trap is "Arrivederci;" while the phrase is used as "goodbye" in Italian, it literally translates to "Until we meet again." And Ratigan and Basil indeed DO meet again...after Basil manages to overcome his funk, get himself and the doctor out of the trap, and thwart the professor's evil plan.
While it's never outright stated, it's heavily implied Basil idolizes Sherlock Holmes; not only does he clearly model himself after the human detective, his own home is located within the walls of Baker Street. This is in stark contrast to Ratigan, who shows no such fondness for Moriarty despite their obvious similarities. And this makes perfect sense, because just like James Moriarty, Ratigan is a psychotic monster who doesn't care about anything other than himself and proving himself superior to Basil.
There's more to the onslaught of Basil's Heroic BSoD right before the climax than just Ratigan outsmarting him again. While understandably frustrated at the beginning when the ballistic evidence doesn't line up and when Olivia gets captured by Fidget at the toy shop, Basil is back on the case as soon as a new lead (such as Olivia's mentioning that her father had been kidnapped by a bat or Dawson finding the list) comes up. So what makes this time different? It's not just Ratigan proclaiming that he's won. It's Ratigan, Fidget and the rest of the minions laughing at him. Why? Remember, Basil has shown touches of arrogance. What is arrogance often a mask for? Believe it or not, insecurity. Deep down, Basil doesn't just fear being outsmarted. It's everyone seeing he's been outsmarted and mocking him because of it. That's why he completely crumples this time as opposed to the first two times around Olivia and Dawson (who didn't laugh at him); his worst fear has literally come true.
Felicia, Ratigan's pet cat, is very fat. And given that she seems to dine exclusively on mice given to her by Ratigan...and the reactions of terror and fear in the henchmen when Ratigan pulls out his signature bell certainly speaks to that too.
The one time she's shown eating a victim, she deliberately swallows him whole and alive. Death by Felicia definitely sounds neither quick nor painless.
During the bar fight, several mice are shown lying motionless. Since several of the patrons had guns, knives and other lethal weapons, were these mice KILLED?
It was followed upon… sort of. There was an awesome comic, never printed outside of Germany and northern continental Europe, which was about Ratigan's return. The comic had a page explaining how using his cape as a parachute, combined with his light weight and the luck to fall on the soft, velvet-covered roof of a passing carriage, allowed the villain to stay alive.
During Ratigan's first official appearance, he tells Flaversham "You know what will happen if you fail" while holding the bell, and Flaversham reacts to the bell with noticeable fear, even more than what he displayed upon Ratigan's previous threats. Seeing how he later uses Bartholomew's death to enforce that he isn't a rat... chances are good he's given Flaversham a little demonstration regarding Felicia and that bell before.
When Ratigan details his changes under the guise of serving the queen, one of his changes is taxing the elderly, infirmed and children. Hiriam and Olivia fit the first and last. In essence, Ratigan was never actually planning to let the Flavershams free, and would keep antagonizing them in some way even after completing his coup.
After Fidget is sent to get the items on Ratigan's list (including Olivia), he manages to find her with relative speed and ease in the safety of Baker Street (although he is unable to capture her right at that moment). Given that Ratigan is established as someone with enough resources and loyal minions to plot an (almost) successful coup, Fidget probably talked to his contacts and got a good idea about where Olivia was and how to find her...
