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I don't refrigerate eggs for my family's use, if I know they are going to be used up within a week. After a week, I pop them into the frig. The US is one of just a few countries where folks routinely refrigerate their eggs. In many parts of Europe, eggs are sold on the grocery store shelves, just like canned goods.
Eggs do age faster left out on the counter. This can come in handy if you plan to hardboil the eggs, as really fresh eggs don't peel well once hard boiled.
To the OP - I don't wash the majority of the eggs I use or sell. The occasional dirty egg I scrub clean with a soft kitchen scrubbie I keep for that purpose, polish with a paper towel and in the cartons they go. More than a little dirt and I will either wash and put it in the "use first" carton or feed to the dogs.
Wash in water at least 20 degrees warmer than the egg itself, dry immediately and refrigerate. No detergent is necessary and really does nothing for the egg.
IMO, it's best to leave the bloom intact whenever possible.