The dog rules with an iron fist (paw?), so have to say no.
Your dog' s tail is wagging because he is exited or likes something you did.
The possessive form is: The wet dog's tail is wagging.
It can, but this is more likely when the dog is sitting or lying down. When a dog is standing and wagging its tail, it's the momentum of the tail wagging that spreads to the rest of the dog's body and causes the rest of the dog to wag too.
wagging
Wagging
Its depressed. It wants to be a proper dog.
Ears will be up and tail will be wagging
The farrow hound dog
The Wagging Tail
Imo oten
it probably depends on how fast the dog is wagging its tail
Sniffing, tail wagging, and play bows.