These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Aunt Sarah, especially to people who aren't dog lovers. It's possible she's just prejudiced against dogs and doesn't like them. It's also possible she didn't want to allow any animals near the baby, since she did leave her cats downstairs.
The alligator in the zoo. Trying to eat Lady or trying to help remove the muzzle - and simply not realising his mouth was too big.
Designated Villain: The dog catcher is perhaps the least evil of the Big Bad Ensemble; his job is to find dogs and put them in his pound to await pickup, but the film generally portrays him as antagonistic for this, particularly when he agrees to have Tramp put to sleep. Also, though she may be particularly harsh towards Lady and the Tramp, Aunt Sarah does what she does out of concern for others, particularly for her cats (who get Lady muzzled with a Wounded Gazelle Gambit) and for Darling's child. Plus, she hates runaways (and also, it would seem, strays), to the point of chaining Lady to the backyard doghouse to teach her a lesson, and around the time she arranges for Tramp's euthanasia, she's only aware of two things as far as the surrounding circumstances were concerned: one, that the child's life was in danger, and two, that two dogs, one of them a stray, were at the scene.
Ethnic Scrappy: The infamous Siamese Cat duo. On top of being, of course, villains, their Ethnic Scrappy status manifests itself in their buck teeth and the annoyingly terrible grammar during their song.
Fair for Its Day: The ethnic stereotypes personified in animals, the cats in particular.
Memetic Mutation: "I SAID A LOOOOOOOOG PULLEEEEEEEEEEEEER!!!!!!!!!!!!!!!!!!!!!!!!!"
Surely, the beaver counts as well, since, despite only appearing once, he still has a fairly significant role, and is very helpful and friendly. It also helps that he was the basis for Gopher.
Squick: As the entry on the "Honorable Marriage Proposal" page mentioned, it's possible the scene where Lady is being chased by the dogs is caused by her being in heat, as later on she spends a night alone with Tramp, and Jock and Trusty discuss marrying her to preserve her honour.
Villain Has a Point: Aunt Sarah putting a muzzle on Lady. It's generally not a good idea to allow dogs near newborn children and Aunt Sarah believed Lady was violent and destructive. She was doing what she thought was best for the safety of her nephew.