I do not wash mine and they sit on my counter. In cool weather they are good for about 6 weeks. Mine never last that long at all. We eat them too fast!
I know that in Europe they never regrigerate eggs, I think alot of countries are like that.
I don't know how it got started that they had to be refrigerated here. It may have been that thing where it makes them last longer if you refrigerate them and then it evolved into they will go bad if you don't stick them in the frig.
You know how things progress like that.
Think about back in the day when they didnt have fridges to put eggs in?
I usually do the same with my eggs as everyone else, put them on the counter, they dont last long, I do wash them though, just because some can be pretty dirty LOL
The eggs I get from the store I always keep in the fridge, those eggs are older then most people think when buying from a store.