We have some of these flat egg carton things and they work really well to organize our eggs in.
(Those aren't my eggs, but just a picture I found of the egg flats in use)
We add eggs in the same order one reads in English: top(back) to bottom (front), left to right. Once a flat fills up, we start another, also having that go in the same order. If we have less than a flat of eggs at a time, I generally shift all the eggs every day, so that the oldest egg is in the "1" position, rather than there being a big gap from the eggs we took. Once a flat fills up, we put the new flat on top and start loading it up, making sure to take our eggs from the BOTTOM flat when we sell/eat them.
There's been a few times that we've had more than 2 flats full of eggs . At that point, usually the newest eggs are on the top, with the oldest eggs below them, then the next oldest eggs below them, and the next oldest below them. Once the "newest" egg flat is full, it goes to the bottom of the pile, and a new flat is started. I find these flats really useful for organizeing the eggs, as they stack well and I can tell which eggs are which really easily.
As for where I store them, they get stored in our garage, unwashed. We've found that unwashed duck eggs will last months at 50-65 degree F tempuratures.
If I have some eggs that are dirtier than others, I'll eat those first, regardless of whether or not they're the oldest. I figure the poopy ones will degrade faster. Since I have duck eggs and we sell them generally unwashed, I usually will pull out and fill a egg carton with the cleanest eggs as I get them, and once that carton is full, we sell it.