There was a fairly long debate on this topic over the winter - you can search for that thread, but I believe the general concensus was 50/50 between rings and no rings. Someone contacted Michigan Stinger, who said that they design their spoons to be run without rings, while other guys like the added protection provided by the more free swinging nature of the spoon on a ring, and/or additional action by using a ring and a swivel. This long and short is, it comes down to personal preference. If I buy a spoon with a ring on it, I run it with a ring, if not, I don't.
As far as running a spoon behind a spinny, I personally don't do it - but I would imagine there is a lot of line twist going on there, and swivels will help cut down on it. FWIW, it may be worth running a barrel swivel between the spinny and spoon as well.
Chris