Once you refrigerate them, they must continue to be refrigerated. Refrigerated, they will keep for months, but who would want to? I keep mine on the counter, but they are usually eaten or given away within a couple of days.
I don't wash mine - the nest boxes are clean. If I get a dirty one, I'll wash it and use it right away. Washing removes the protective "bloom" so I don't keep washed eggs on the counter (I refrigerate them) and I use them promptly.
If I want to hard boil them, I will keep them on the counter for a couple of days, then refrigerate for a week to give the membrane time to outgas and shrink away from the shell a bit to make peeling the hard boiled egg easy - or even possible! Fresh eggs won't peel easily when hard boiled.