So what does each pet do? Descriptions and screenshots after the break!
Sits and sleeps. Not sure if it's dependent on player state, but for reference, he sat when I was sitting and he lay down when I was laying down. :)
Creates a neat red portal when summoned. Every now and then, instead of summoning a white imp, Kil'rek (from the Illhoof encounter) will jump out of the portal and follow you instead!
Hellfires when idle.
Spins around while breathing fire.
Summons a Wandering Spirit who, appropriately, wanders around. :P
Will gain the buff "Purified" and turn blue for a short period of time.
Farts a bubble and then pops it (same as its larger counterpart's animation).
Emits a small aoe and causes nearby critters to run in fear.
Summons a tornado (cows and farmhouse included!) that also picks him up.
There were two pets that didn't really have any animation (that I noticed). The Menagerie Custodian doesn't do anything, but he does quote the Curator from time to time. And the Pocket Reaver didn't do anything or say anything at all, other than the usual motorcycle revving gesture. If you've found any unique animations for these two, please let me know! I'd love to see them for myself :)