When I use to sell duck eggs, I made nests out of cat carriers. Each carrier makes two nests. When I collected the eggs in the morning, I cleaned any nest that was in need of it. Dave Holderread told me that most ducks lay their eggs before 9am in the morning. My experience confirms this. I kept the ducks in their duck house with their nest boxes until 9am at which time I would let them out, collect eggs, clean nests.
Some ducks did not like to use the nests, liking to be more sneaky and hide behind things. For these ducks I used plywood laid up against the wall at an angle with just enough room for a duck to squeeze in there. There were quite a few duck who preferred that. I also found these the easiest to clean because the ducks rarely slept in there, only went there to lay eggs, unlike the nests.
Of course there were some dirty eggs. For those eggs that didn't just rinse off, I used a scrubby.
And yes, you can refrigerate an egg after washing. It's recommended.
jofanx asks a good question about where the nests are located. That can have a lot to do with how clean they are.