Have you checked your nest boxes recently? If they have become fouled, it's time to strip out the bedding and freshen things up. If the nests are really poopy it is more likely an issue of someone using them for a residence rather than a place of business, and that is something to be addressed separately with some good roost training, etc (most of the time, a bird that is using them outside of laying will do so for sleeping rather than going to roost). If the nests are just a bit messy it may be that you have a few girls who's "valve" isn't fully effective in shutting things off at the time of passing her egg - it happens and sometimes some poop gets through with all the pushing to expel the egg. Not really much you can do about that, though if you are noticing some looser than desired poops going on in general it may be something to consider trying to fix (once you determine the issue - could be parasite infestation, feed issues, illness, etc.) as the better the poop the less likely accidental leakage is to occur. Another common cause of a nasty nest is hens tracking stuff in on their feet when they go to lay - they move around the nest, walking on the eggs already there and, after, their own - leaving the stuff from their feet on the eggs. If this is the case it's more a matter of addressing the problem from a management point to cut the amount of loose, muddy, mucky stuff the hens are getting on their feet - which may mean attention to your coop and/or run area.
ETA - the other questions - any egg that has been washed should be refrigerated (noting in case you are one that prefers to keep eggs on the counter). Use the least amount of washing you can for the egg in question - you may be able to dry wipe the waste off, use a damp cloth or, for really stubborn eggs, do a full wash with plain ol' water.
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