Sorry for sounding ignorant, I'm still getting the hang of being here.
I did read the FAQ and it said that pets, indeed, can't die. However, I just wanted to be sure since games with feeding mechanics usually let pets die anyway after a while.
Nope, pets will never die.
As it is, I think the only actual consequence of having 0/10 hunger or happiness is that the pet can't battle.
Pets can't die. The only consequence of not feeding them is they're ineligible for battles.

as already said they can't die but it's very easy to keep them fed and happy, just put them in the Veta Lake Resort (or in the cheaper Rathskeller Inn if you have access to the islands) :D
Pets on neopets don't die from starvation either? So it's not everywhere that this might happen. Subeta's more relaxed than those other places, so no worries.
There is no disease mechanism here, all the pills/bandages at Medicontra are for people. Your pet can get totally stomped by "the big bad" during collective battles as part of huge Sitewide Events, but will stay alive just fine. If you want them to have hit -points just for the lulz, NPC Selena will heal them at minimal cost.