[DoctorDraca]

I don't know whether you've read the front page or not but when there's no dot, it means the dog is taken by someone and you cannot apply for it.

Custom pets that members hand out in the oekaki have nothing to do with the official CS pets. The CS pets are made by the staff, while the pets in the oekaki are made by members. CS pets can be collected and stored on your account, while these creations can be used as characters.
The ones that people hand out in the oekaki are self-made creatures (sometimes based on existing species) which people can "adopt" and treat them as their own characters with their own backstory, life, etc. I don't know whether I'm making myself clear though.