PlayAnimation reset the animation queue and plays the animation you specify.
PushAnimation push the animation at the end of the queue. Which means it will wait for all the previous animatios in the queue to be played before playing the one you just pushed.
I'm not sure about this but the no eyed deers may despawn in summer or spring if there's nobody near, for the volt goats you must move one and kill the rest of the herd, BUT! I think you need to move it without teleportation, a friend told me about that, I never stole the whole herd before, I don't know if actually teleporting one goat may work, but I'm sure you need to kill all goats in the desert.
Finally for the catcoon, go to the critter station and craft a kitty you will get a invulnerable pet, if you want an adult make fences, they don't starve I guess, just be careful if you with glommer