You don't have to refrigerate eggs at all, if your house isn't overly warm. In England and Germany, they sell eggs on the shelf next to the bread. I like to refrigerate my eggs, but I had a friend who always kept hers on the kitchen counter, and she gave me quite a few - I never got a rotten egg from her.
You can wash eggs with a mild soap, but it washes the "bloom" off of it. The bloom is a naturally occurring protective coating that is on the eggs when laid, and with it on no air can get into the tiny little pores on the egg. They will stay fresh much longer than washed eggs. I don't wash my eggs unless there's poo or something nasty on them.