I think a great area to rent in Pasadena that meet all of the qualifications you listed is near Old Town, specifically Marengo Ave has a lot of great rentals in various price ranges. Marengo can be a busy street, so if you're looking for something a bit quieter there are also a lot of great options surrounding just East of Marengo Ave. Many rental complexes in this area have subterranean garages, and a decent amount are pet friendly (depending on your pet!). Old Town is also a great area to shop and discover new restaurants. Let me know if you would like more information!
I would be happy to help you with your search. Do you want a single family home or would you consider a townhome with a small outdoor area, as well? There are many options in the Pasadena area, you can reach me at (626)487-3678 to discuss the matter in more detail.