I've seen it a few times now, but I haven't figured out how I can add my pet's headshot under my avatar. How do you do it?

And, while on that topic, will it change according to which pet is active, or is it one specific pet?

that's the pet leash feature, which is only available to gold accounts. you can pick any one of your pets to display beneath your avatar, and switch to a different one at any time.

Ah. Thank you for your answer.