I agree, why not let our characters and pets do all of the above since it would bring life and joy to our pet, characters and most importantly the players.
Btw I kinda like it if my pet wont just follow me around or keep on circling around me. Seriously it should be dizzy or tired from doing that all day long.
Instead do something like sit when its tired, making faces when its hungry, poop when it needs to, play dead when the characters are dead so it could save itself and returned to us when we revived, and etc.
Oh and for the characters maybe, sitting down, sleeping, stretching, yawning and etc.