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.
Lady looks extra shocked when she initially meets Bull in the pound. Of course she'd be- she's been around purebreeds her whole life, yet finds that most of her cellmates are purebreeds: a Bulldog, Pekingese, Russian Wolfhound and Chihuahua. Lady probably thought the pound only had stray mutts like Tramp.
Also, the reason poor Lady is so distraught when she shares her experience at the pound with Tramp is because she second-hand witnessed a dog being taken to be put down, the equivalent of watching a man being sent to the chair. Of course her experience at the dog pound was horrible. It goes against everything she and Trusty and Jock believe in humans being "dog's best friend".
Tramp has a reputation for having a lot of previous "girlfriends," a statement a child would take at face value, while an adult would understand that he was a stray who impregnated a lot of neighborhood dogs. In short, he reallyiskind of a tramp.
Tramp's impression of what happens when a baby arrives is funny in some points. However, where did he get this knowledge?
"The voice of experience buster—move over."
I took it that he was already owned at some point or, to the least, knew some dogs that were and were thrown out when a new baby arrived, which seemed common back then.
The scene with Lady in the doghouse, and Jock and Trusty come over to offer some comfort. Younger audiences will see it as the two offering to take Lady into their homes, where she'll be away from Aunt Sarah and treated well; older audiences will realize that Jock and Trusty were actually offering to marry Lady in order to preserve her honor, especially with the implication that she had become pregnant after her night with Tramp, AND with the knowledge that he was a womanizer. The movie is set in a time period when it was considered shameful for a woman to be an unwed mother, and the only way to save her reputation was if a man of high social standing agreed to marry her.
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 have a friend take care of her temporarily. Then he wraps the gift and says "Oh Darling, let's open a present".
In the movie proper, there is no indication that the family was aware of Tramp, so this issue is nonexistent in the the film itself. On an old interactive book-on-tape, however (a fully licensed Disney product), Jim Dear refers to Tramp by name immediately after the rat is discovered. So, you are aware of a stray dog in the area who apparently has a reputation for many forms of mischief, and you don't at the very least think to get your little princess of a pooch fixed?
In all fairness, spaying and neutering at the time wasn't really well known, let alone commonplace, along with Lady being so obedient, usually.
Spaying female dogs didnt become functionally possible until the 1930s, and wasnt remotely common until the 60s-70s.
Trusty and Jock run down the dogcatcher...who doesn't, you know, stop so he can catch them, a couple of dogs harassing traffic.
In fairness, judging by the dogcatcher's lines and the speed of the horses—nearly a flat-out gallop on city streets in the dark?—it's made fairly explicit that Trusty's baying and snapping have thrown the horses into a panic. The dogcatcher is trying to stop.
What is Aunt Sarah feeding that infant? Infant formula existed in the movies era but it was fairly crude and wasnt something youd chose unless necessary following the mothers death or abandonment or similar.
Possibly goat milk, which is about the closest to human milk in terms of taste and digestibility that you could get before modern baby formula.
Why would Jim Dear and Darling leave their newborn baby at home? Even if they leave their aunt to look after him, it doesn't seem like a very good idea for a mother to leave a baby that young. Also, who was looking after the baby when Aunt Sarah wasn't there (when she went to the pet store to buy a muzzle)?