Simple question, can they die? Not being able to buy food in bulk means that buying an item 10 times just to make them not hungry is really rather annoying, I know about the resort but still, can they die?

No, they can't and they also don't run away or something like that. A hungry Pet can't battle though so if you want to visit the Battle Arena you should make sure that at least your Battle Pet is fed and happy.

Awesome, thanks for the answer :)