Also on a 737-800 do the engines operate in the same direction?
Yes. I'm not aware of any jet aircraft where the engines are 'handed'. On turboprops the engines themselves are not 'handed', but gearboxes are. Not sure about the new geared turbofans, but I see no reason for them to go down the turboprop route.
As for your first question, do you mean a single engine aircraft with contra rotating props, or twins with counter rotating props on opposite engines?