If you wash the eggs absolutely put them in the fridge. Washing removes the bloom -- the egg's natural protective coating -- and compromises it's ability to "keep" for extended periods. That said, even if you plan to refrigerate, there's not really any good reason to wash the eggs. Even in the fridge the bloom is helpful in preservation and if you keep clean nest boxes you shouldn't have any problem with eggs being "dirty".
As for keeping them out, people do so for different reasons but as long as the eggs will be used in the next couple of days its really of no consequence. Some people do so as its tradition, sometimes we leave some out on the counter when the fridge is packed to the brim (which happens often in the summer harvest season), some people like the looks of a bowl of eggs on the counter, etc, etc. The reasons are probably as numerous as the people who keep them.