I do not wash mine unless they're visibly dirty and keep them on an egg-tree (Not sure that's what it's called...) on the kitchen counter. If they do need to be washed, then they should be stored in the fridge afterward.
Store them however you feel comfortable. I refrigerate as I cannot possibly eat up eggs fast enough to keep them on the counter (they "age" faster at room temperature, vs refrigerated).
If you wash them they must be refrigerated, as washing removes the bloom which serves as an antibacterial barrier.