Q: What is the best way to wash and store my eggs?
A: It's counter-intuitive but true: Your eggs will stay fresher if you don't wash them at all. When your hens lay eggs, there is a natural coating that is laid on top called the "bloom" that helps keep out bacteria. When you wash eggs, you drive some bacteria in through the pores of the shell, so it's a bad idea to do so as a general practice. If your nests are clean, your eggs should be clean. In fact, fresh eggs don't really even need to be refrigerated if they're going to be used soon. They can be kept at room temperature, although refrigeration will keep them fresher longer. (We always refrigerate our eggs--it just makes sense to keep them as fresh as possible!) Store them large end up.
A hen lays an egg a day, and after 10 or 12 days or so, she has gathered a clutch together and will begin sitting on them to hatch them. The first egg she laid has been sitting in the nest for two weeks or so, but is still good enough to turn into a baby chick!
Commercial eggs must be sanitized because they are often laid on top of feces---or even worse.
If your eggs are as soiled as that for some reason, some people recommend you use sand, sanding sponges or sandpaper to carefully buff off the dirt... but it's really best not to wash them in water as that can spread the bacteria to the inside the eggshell. If you are determined to wash your eggs no matter what, your solution should be slightly warmer than the egg you are washing to reduce the amount of bacteria you are driving in (as the egg cools, what is on the shell will be drawn inside, and if you have removed the bloom, the bacteria will enter). It seems like a waste, but if an egg is very soiled or caked with more than just surface dirt, sometimes it's not a bad idea just to toss it out.
Keep in mind that if you are consistently getting very soiled eggs, you should consider changing your flock management practices. Check to see if someone is ill, and that's causing loose droppings. Determine if your hens are sleeping in the nest boxes (and pooping there!) rather than on their roosts. Maybe you will need to change the height of the nests or the roosts, or darken the nests so they will be less inclined to do anything in there other than lay eggs.