Pets have a mind of their own

Pets should have more of a mind of their own. They should be able to sit and lay down on anything and at anytime they want. They should have multiple ways to lay down, like on their back, or on their belly with their legs sprawled out, or on their side. They should be able to take naps too. Pets could also be playful, like maybe they run in circles around you at random times, or simply just run and jump and roll around. 

Pets could also have randomized personalities, like maybe one pet sleeps a lot and doesn't move very fast; one pet is super hyper and runs and jumps a lot; one could be really shy and hide behind you when people or mobs come close to it; one could be temperamental and attack every hostile mob and growl at other mobs and people; one could be protective and never leave your side, and doesn't do anything without cues from you. Of course there could be even more personalities.

P.S. - This should go for every tamable pet: wolves, cats, parrots, llamas, horses, donkeys, mules, etc