After speaking to Katina about my problem, she suggested I try writing a storyboard for the different activities the pet would be able to do, and then use the timeline so that the user will be able to control which activity the pet will do and when.
I would also like to be able to create a "happiness-ometer". The pet's mood will change depending on what the user is doing with it (e.g. if the pet is being fed, happiness will increase, and if the pet is being hurt or neglected, happiness will decrease). If the pet's happiness decreases to a certain point (lets say zero), he will run away.