Because I am selling most of my eggs to those folks around town who want fresh ones, I feel the need to wash them before I put them in the frig. The reason being, it is spring and muddy and poopy all over here and in their coop that sets on the ground. Once my new building is done and they are going to be having many more cleaner boxes and roosts to use, I don' think I will have as much need to do so. As long as they are dirty/poopy now, I will wash with water about 100-110 degrees and then a short (10 seconds or so) in a bleach rinse that has about a capful of bleach in a gallon of water, then set them to dry before putting in the carton for frig time. This was how the State recommends clean for hatching as well.
At the very least, I rinse under running water at that temp and rub off all the mess I can. Sometimes the shell still has some slight stain. (yes, I have some who sleep in their boxes, but I am working on that).
As for peeling, I had cooked up some hard boiled eggs just a couple of weeks ago and I had no problem peeling and they were that days eggs. I put them in the cold water in the pan, then I turn on the heat. I bring it to a rapid boil for 1-2 minutes, then leave them on the stove in the hot water for 20-minutes. The peeled like a dream! I even showed my son how if you crack the shell all over, peel open both ends about the size of a quarter, you can actually blow them out of the shell. I couldn't believe how good these eggs peeled. I usually have a mess on my hands trying to get the shell off the white it is sticking to.
I have tried using vinegar in the water before too and it didn't make any difference.