I'ld say don't wash, as that destroys the bloom on the egg - a sort of wax that protects the egg. If necessary, I wipe them before using them. I keep mine in a wire egg basket on the counter, and don't refrigerate them. I think they would keep a good month that way. After a couple of weeks, I do put them in the frig, where they are good for another couple of months if you need to keep that long. I've planned ahead for egg shortages that way, by stockpiling.
I'm sure some panicked folks will warn you to keep them refrigerated - but in 8 years I've never had a problem, even in hot weather.