I don't wash them, and I don't refrigerate them. the bloom helps keep them air tight and keeps them from aging or spoiling, and they don't need refrigeration to stay fresh. if they're particularly icky or have stuff on them I don't want dropping into the bowl when I crack them, or if I'm hardboiling and they're icky, I wash right before I crack them or boil them.
I did the great egg experiment a few years ago, unwashed duck eggs will keep on the counter for 4 months or more and still be quite usable. (ok, you do lose a few to spoilage, but trust me, you won't use those by mistake!)
note that once you refrigerate them, you have to keep them refrigerated... eggs that were refrigerated and then stored on the counter are likely to spoil.