I use "born-on" dating, and mark each egg with the date I collect it (in pencil).
It helps me know how long they have been in the basket on the counter (leaving me some to scramble or whip easily at room temperature), and when I have a dozen, I put them in the fridge.
My family and friends absolutely love seeing the date they were layed! They get such a kick out of it!
I try never, ever to have to wash my eggs. My nests are clean and if something happens to stick to it, a quick swipe with a paper towel usually does the trick.
If one is really, especially gross outside, I will cook it right away for the dog or the chooks.