Ushi No Tane Forum

If I'm unable to buy more food for my animals because it's a rainy day, how long can they go without it as long as I heal them later?

If it's raining both today and tomorrow, and you are out of food today, I would reload my game until tomorrow is a sunny/half sunny day, because your animals will lose FP:
http://fogu.com/hm10/your_farms/animal_friendship.php

Pretty sure you can keep track of their health by checking the animal stats. I'm not sure whether they die or just get sick once the blue stress bar is full, nor am I sure whether you're notified once your animals are sick. Maybe that's what the Stethoscope is for? I've only had one animal die on me and it was due to old age. If you do let them die from sickness you'll lose FP not only with all your animals but the villagers as well (it's a pretty nice chunk, too, so you'll want to avoid it if you can).

They'll start to become stressed, then they'll get sick. Only after becoming sick they'll die.