First a quick and cheeky disclaimer: all of the following information comes from my own 20+ years of experience raising chickens, eating eggs, talking to people, reading, and thinking about things in a critical way, but I am not infallible. Much of my statements may contradict established food-safety guidelines, and so I feel compelled to say that while I strongly believe everything I say to be utterly true and factual--and moreover, scientifically verifiable--I may be wrong, and so your choices and any consequences must remain your own. If this scares you, feel free to continue to let other people think for you. If you are okay with accepting responsibility for yourself, then by all means, please read on!
Well, the FDA, USDA, or the egg industry won’t tell you any of this, but basically, the eggs are in their cleanest, purest, safest state just the way they are when they pop out of the hen into a clean nest. Each egg has a natural powdery coating, the “bloom,” which protects it while still allowing it to breath. Since egg shells are porous, you don’t want to wash them–washing makes the outside LOOK clean but destroys the bloom and can (theoretically) allow foreign matter, poop, or pathogens to enter, and speed evaporation of the liquid inside.
By the way, fresh eggs in such a natural state don’t need to be refrigerated either–they are designed by nature to last at room temp for at least a couple of weeks (FYI fresher eggs are generally preferred for frying while older eggs have a runnier texture but conveniently peel easier when boiled). Remember that egg farming is much older than refrigeration, but refrigeration can extend the “shelf life” a bit if you need it–however, room temp eggs, unsurprisingly, work better for most recipes so most people will leave cold eggs out to warm up before cooking.
Tests have shown that unwashed eggs keep longer than washed eggs. Personally, I never, ever wash our eggs, nor refrigerate them either. Keep your nesting material in your boxes as clean and dry as you can, within reason. In the occasional event that you get a badly soiled egg, you can wipe or wash it off if it bothers you, but this does more to make it look presentable than anything else–you won’t make the INSIDE cleaner, or make the egg last longer. You can use up the very dirty eggs first, since they may not keep as well. If you every doubt the integrity of an egg, you can float it in water (if it floats, it’s bad, they say), or crack it into a cup or bowl and give it the “look and sniff” test (you’ll KNOW if it’s bad, trust me). Basically, don’t panic–it’s organic!
As to Salmonella bacteria and eggs, in a nutshell, my unprofessional but somewhat educated opinion is that the chances of home-produced or small-farm eggs–from ostensibly healthy birds, raised appropriately–containing Salmonella, or at least any harmful strains, are generally remote enough to be negligible. The prevalent, naturally occurring varieties among birds tend overwhelmingly to be more or less benign. The mutant, virulent strains we hear about from factory farms are a novel phenomenon and are closely associated with the many abusive practices of that system–NOT with eggs and poultry per se. So feel free to enjoy your homemade mayonnaise and poached eggs from your happy hens!