How can I get another pet? I would like to find out asap, as I have a name that nobody has yet.

You'll need the mystical donodak or montre plushies, or a specific totem from the loyalty box. Barring that, you'll have to abandon one and create a new pet.

or both give you extra pet slots.

When your account is two years old, and then ever year after that, you get a loyalty box and you can use the Loyalty Pet Slot Token to get another pet slot.

Thank you very much. Is there a way to change the name of one of my pets?

allows you to change the name of your pets.