My 5 year old cat does this most mornings. He hangs his head way over the side of the cat tower or from another cat cubby we have, and appears to be sleeping. He doesn't walk around with a dropped head. He is normal for the rest of the day. Is this just a quirk?
Yup. That's what it is, and a LOT of them, if not all, do it at some point or another. Don't ask me why. They just do. Part of their general cattiness. Hekitty (see avatar) not only hangs her head over stuff, she occasionally "oozes" off of whatever she is lying on. Doesn't even wake her up all the time. Gotta love the kittehs!