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.