Yes, my cats. Especially during the winter when it's cold. They want to be under the blankets and will come sit on my pillow and tap my shoulder or head until I wake up and lift the blanket for them... Spoiled brats. ;-)
My dog, Gretzky, will stare at me till I wake up in the morning
Not mine. It's usually feral cats fighting outside my window or coyotes.
Rarely. Mine actually helps me get to sleep. There is something about having my hand resting on my cat that helps me feel peaceful enough to finally fall asleep.
She'll probably mature out of it. :)