How long was Lady kept in that gift box before Darling opened it?
Doesn't have to be too long. He could have bought Lady that evening and kept her hidden in the garage - or had a friend take care of her temporarily. Then he wraps the gift and says "oh Darling, let's open a present".
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.
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.
Tramp being falsely condemned as a threat to the baby, only to be exonerated when the dead rat is found, is actually pretty close to the French tale of "Saint" Guinefort, a medieval greyhound venerated around Lyon until the 1930s. In that account, the dog kills a snake in defense of a nobleman's baby son, but is killed by its master when the man finds the boy's cot overturned and the dog with bloodied jaws. Finding the dead snake and the unharmed child moments later, the repentant knight erects a shrine to his unjustly-slain pet, and local peasants who visit it to praise the unfortunate Heroic Dog soon start reporting miracles akin to those of human saints.