Remember how Lady's owners were referred to as Jim Dear and Darling? (Like even in associated books, etc...) If you also thought "well that's sort of a strange name...", it makes sense when you realize that those aren't their real names — they're what they call each other. As Lady knows them from only the privacy of their shared home, it makes sense that she would hear only these terms of endearment as "names" and probably assume that is what their names are.
Tramp is a mutt, but he looks like he's got a lot of terrier in him. Terriers are rat catchers.
Lady being cold to Tramp at first then hoping Tramp didn't leave seems odd at first. Then, you realize that Lady is already pregnant with her and Tramp's puppies. She was experiencing moodiness.
Her anger at Tramp having past flings takes on new light when you realise she was raised by Darling and Jim Dear - she grew up assuming that people only ever had one pairing. She thought First Girl Wins.
Aunt Sarah believing Lady to be responsible for the mess downstairs that the cats caused. Besides their Wounded Gazelle Gambit, she had good reason to believe it was Lady. The main one being that Siamese cats don't behave like that in real life.