I'm from the UK and I don't know a single person who refrigerates their eggs... we always keep them out and none of us have been ill haha. Mind you it might be down to the difference in temperature and humidity, it's usually cool here, we hardly ever get warm weather. When you buy eggs in supermarkets here they are not refrigerated!
It depends on how the egg went through processing. Eggs that are fresh off the farm and usually bought directly from farms do not need to be refrigerated. Eggs that are typically bought from the store need to be refrigerated, because they've had a layer of natural protection washed off for cleanliness concerns for consumers.
I don't think you have to refrigerate them, especially if they are straight from the farm, but they go bad pretty quickly if you don't. But I would say if you BOUGHT them refrigerated, then you should keep them refrigerated.
Eggcelent question! The eggshell is a great barrier against bacteria, however, what if there's a crack and you dont notice? Furthermore, if eggs aren't meant to be refrigerated then why do we have egg holder on fridge doors?
Yes you should :P Also don't wash them before you crack the eggs , it'll dissolve the protective membrane between the outer shell and inside and drive the bacterial and all sort of gunk inside the egg. Old wife tales aren't always accurate.
Various bacterial such as salmonella that can potentially can cause food poisoning exist on the shells of eggs to some degree... By refrigerating the eggs bellow about 50 F, you slow the growth of said bacteria and thus reduce the risk of potential sickness. It's wise to refrigerate your eggs, but eggs will last a couple of weeks refrigerated if fresh.