If they're not dirty, they don't need washing at all.
My version of cleaning has nothing to do with removing germs--it's just to remove poop and other physical stuff that might get into my food. I trust thorough cooking of the eggs, and washing of my hands, to deal with germs.
I use my hand to brush off anything loose (like bits of nest material); I ignore minor smudges; and then I use plain water on the ones that are still dirty enough to bother me.
I either wipe with a wet cloth, or run the eggs under water, but don't put them in a container of water to soak. Usual advice is that the water should be warmer than the egg (more comfortable for the hands, too.)
If the nestboxes have clean bedding in them, I usually find the eggs are clean enough for me.
Storing: if you don't want to hatch them, an egg carton in the refrigerator is fine.
How long? Well, someone ran an experiment and said the eggs were still good to eat after 7 months in the fridge!
Here's my source for that:
https://www.motherearthnews.com/real-food/how-to-store-fresh-eggs-zmaz77ndzgoe
I had to click the "continue" button, then scroll way down to get to the conclusions section.
They also said that eggs stored on the counter (65-70 degrees) were still fine after a month, probably still edible after two months, but clearly rotten by four months.
Personally, I tend to write the date in pencil on each egg, when I collect it, and then I can be sure to eat the oldest eggs first. I've never yet had an egg go bad in the refrigerator, but I also don't stockpile giant amounts of eggs, so I doubt I've ever stored them for very many months.