Keep collecting other pet food via pet capsules to continue leveling up your pets. Pet supply boxes are awarded every 5 levels (ergo, level 5, level 10, level 15...). It takes a lot of time (unless you're willing to spend a large amount of ECs), but it's a good way to practice patience.
Yup! The pet supply boxes earned from a specific pet only give items from other pet families. For example, leveling up a feline and earning a pet supply box with give you items for canines, Avians, and pocket pets. Earning a pet supply box from an Avian will give items for canines, feline, and pocket pets. And so on.