I know what some of the pets do:
Monkey = dance
Sylph = punch the air and twirl
Rabbit = wash face (I think)
Attack pets do different things depending on whether they are under or over level 40. I can't remember them all. Hopefully someone else will finish the list.
Penguin = after 40, does something then flops onto belly with chin in hand
Bear = after 40, does something then sits down (we used to have fun making the bear sit on chars/pets
)
Bird = fancy flying
And if you are really quick with switching between your pets (W key and charming action in your skills bar) you can get both the pickup pet and the attack pet to do their charming action at the same time.
Lol, the things you find out while waiting for a dc'd trader to reconnect.
