Will the wolf pup in Home die if you don't take care of it?

No it won't die but from what I can find out... You want those stats to be high because you are raising a wolf that will be added to your pack. I have one and I'm on Day 13 and I guess they don't hit full grown til Day 50 so I'll update then if I remember.

He won’t die, he a mini game and if u took care of it u receive a lot of presents, but it doesn’t go to your pack