"Doggie paddle" is a type of swimming style where your arms and legs stay under the water and you move them a bit like a dog walking.
You paddle with an oar. If you are talking about paddle wheels, the wheel turns because of the bucket boards. The boards dip in the water and turn.
"Dogies" is a slang term for cattle.
That depends completely on how the man wants to be paddled. Therefore the best person to answer your question is the fellow who is to be paddled. If he is new to being paddled, my suggestion is to...
an engine
without knowing the car, i am assuming it is the turn signal light that is on the outside edge of your rear view mirror.