If a pet is malnourished, do you always have to pay gold to fix it? Is there another way? Would restocking food and waiting X amount of time cure them?

Restocking food will EVENTUALLY fix it, however how food works is Pet 1 (first pet in the stable) eats until they're full, then Pet 2 eats, then Pet 3 and so on.

So say you have 15 units of food.
Pet 1 is at 11/15
Pet 2 is at 9/15
Pet 3 is at 0/15
Pet 4 is at 0/15

In this scenario, Pet 1 will eat 4 food units, putting you at 11 left, then Pet 2 will eat 6 units, putting you at 5 left. Pet 3 will then eat the remaining 5 units and Pet 4 wont get anything.
However in time, all the pets will eventually get full, as the second day Pet 1 & 2 will only need 1 unit, so 3 & 4 will get to eat more.

Thank you :)