Just my two cents and how I see it. Eggs like all living things age. How fast they age depends on the temp of the environment the eggs are in. The cooler the environment the slower the egg will age.
I live in a very warm to downright nasty hot environment so I refrigerate my eggs as soon as I can, but thats just what I like to do. In the winter time I dont worry about it so much. If you are going to leave your eggs out and not refrigerate I would not wash them as washing them will remove the bloom that protects the egg and will cause faster ageing and increase the chance of bacteria entering the egg. If you do have to wash them I would give them a very light coating of oil to help protect them, but that is just me.
I have read studies where, if I can remember, they were saying they had eggs last up to 8 months and were still good to eat. Although, I think the researcher said he did not prefer them.
Eggs are incredible things and they can last a long time as long as they dont get too hot or too cold. ITs just going to come down to what you prefer. Put some in the fridge as soon as you can, leave some out. See what works for you. Do some research on eggs There is a lot of information on the net- and then youll be able decide what will be best for you.