I think there are two separate issues here.
Are you asking how long they will stay fresh (that's what the title says) or how long they will stay safe? Safe and fresh are not the same thing.
Evaporation of internal moisture ages an egg. It loses its freshness through its shell every day. This does not immediately affect its edibilty, but does reflect on its 'freshness'. Fresh eggs are dense with moisture. In a bowl of water a fresh egg sinks to the bottom and lays on its side. The longer it sits on your counter, the more moisture evaporates out and the more air that gets in, the more 'stale' the egg becomes. This egg, dropped in a bowl of water, may bounce a little off the bottom and then the fat end may lift a little. You can test freshness this way. Does the fat end of the egg stand straight up in a bowl of water? THROW IT OUT! Does the whole egg float? THROW IT OUT WITH GREAT CARE as an explosion is imminent!
Putting eggs in a carton in a fridge slows the evaporation. BUt fridges are dry. Uncovered eggs in a fridge may in fact age faster than eggs on a counter. But this is affected by MANY variables: internal humidity of your house, the porosity of the egg shells, some hens lay thick eggs, some hens lay thin shelled eggs. The shelf life of an unrefrigerated egg even changes season to season depending on how you heat your home. So 'fresh' is determined by loss of moisture and influx of air. Airy eggs will float differently when immersed in water.
SAFETY is another issue. If there is any chance that bad bugs are inside your egg, coldness slows down their reproduction. Warmth encourages reproduction. So a mildly poisonous egg at cold temp will become a wildy poisonous egg at room temp. (actually this is just a guess but I think it sounds brilliant) We refrigerate eggs to slow down any potential baddies. It is an insurance policy. Since we cannot guarantee they are bug free, we stick them in the fridge to stack the odds in our favour.
I have had dud eggs sit for a week in the incubator, being HEATED and broken them to find them looking and smelling the same way as the day I put them in. I have had others that go poof when you crack them and wow, the smell! So..an egg is a bit of a gamble, they will survive just fine on the counter, unless they go bad and you have no way of knowing what way your egg might go. They may be FRESHER on the counter (depending) but SAFER in the fridge.