You'll get lots of opinions here on BYC about the best way to handle fresh eggs.
I have a rooster (fertile eggs) and several egg customers. I wash mine in plain hot water, as hot as my hands can stand it. Dry them well and refrigerate. If I didn't have a rooster I probably wouldn't be in such a hurry to get them in the frig.
ETA: I don't wash the eggs that we keep for our own use, unless they are dirty. My customers insist on having theirs clean.
as for how "safe" i dont know but i use just warn water and a wash cloth there is an egg brush link below for nasco http://www.enasco.com/farmandranch/Poultry+Equipment/Egg+Products/Washers+&+Detergents/
and a very nice egg washer i have my eye on i only was the eggs that really need it and those that do not clean up real well are used at home and not sold also my eggs dont stararound very long with in a week after colection they are almost always used up.
I wash my eggs with hot water (it needs to be at least 10-15 degrees hotter than the egg) and a dab of Dawn on a scrubby.
I do the same with those that I set in the incubator. The folks are right, you'll find lots of opinions. I've found there to be no difference in hatch rates from eggs that are washed to those that are not. So I figure why allow anymore bacteria into the incubator than necessary.
And this question from the germaphobe . . I wash mine in hot water in a little plastic container and a little dish detergent, rinse, hot water, dip in another small container filled with hot water and a few drops of bleach. I then dry them and refrigerate. (even wash out egg cartons and dry them that friends save for me) Should I keep these containers seperate, wash them separately or just wash them with my regular dishes? I am totally serious. Any reasons not to wash the "washing containers" in with my regular dishes?