I didn't read a majority of this thread, but in response to the OP:
There aren't dogs and cats. They're not mentioned anywhere. There are other animals around the city. Barakhan, snakes, rats, mice, cockroaches, various beetles and insects of varying sizes. If you look at the jozhal helpfile, it even states that rich people have them:
Jozhal (Animal Life)
These small, desert-dwelling reptiles are hunted for their skin, which,
supple and finely scaled, betrays a multitude of colors. Occasionally,
they are kept by nobles and wealthy merchants as pets, but generally revert
to their feral state if unfed. While not usually dangerous because of their
small size, they are scavengers by nature, occasionally taking smaller live
Or the helpfile for barakhan lizards from the old website:
Barakhan Lizard : These small winged lizards, usually colored in dark or sandy colors, are the usual source of eggs in the Southlands, where they are often domestically raised for that purpose. Some barakhan lizards are kept in cages as pets, since their chirpings can be interpreted by the ear as musical in nature.
There are options out there for people to have pets - be in the shoulder-wearable objects like hawks and lizards and snakes, or something more exotic. If you wanted an NPC pet, it's likely that you could work out a plot for a group of people to go capture you one, then another group of people to domesticate it, all while talking to staff to make an object or NPC for you.