The answer to the question "Why did Princess deliver a Jumping-to-Conclusions Diagnosis in "I Don't Want a Cold"?" is simple: she's a kid. She knows that when she herself caught a cold, her first symptom was sneezing, so a four-year-old could easily jump to the conclusion that sneezing always means they have a cold, hence her saying, "You're all sick!" to the adults despite them appearing fine to someone with experience. And she also did a Jumping-to-Conclusions Diagnosis on Puss because he was coughing. She could easily get the impression that coughing always means you have a cold as the only time the adults knew she was sick was when she started coughing and she didn't know cats were immune to human viruses but did know colds were contagious. The adults on the other hand, only thought the Princess was sick when she started showing proper cold symptoms because, while they're a bit loopy, they're still adults with more experience.
In "I Don't Want Help", when the pets are thirsty, Scruff just stands there panting, while Puss actively tries to turn on the tap. Maybe that's because cats tend to be more independent than dogs.
In "I Don't Want to Go to the Hospital", it's mentioned that the Queen has been there before. Maybe the time she went there was when the Princess was born.
In a way, "I Want My Snail" has a Spoiler Title — you can't want something unless you don't have it; and Speedy the snail dies in the episode.
Fridge Logic
Fridge Horror
The state their kingdom must be in. The four-year-old princess is nearly the most rational person in this castle. The King and the Queen (the same with their personnel) regularly leave their routines to humour the Princess in a frequency that makes you ask how anything is ever going on in politics. The navy consists of paper boats and there is also seemingly no real army (how to defend the nation, just in case?). And where are their people? Well, maybe they just govern their meadow...
