I have a little procedure I use for my eggs: I reach in and pick up the egg. It ALWAYS covered with grime. Then I just take it into the garage sink, where I have a sponge for eggs, and I wash it in the sink with plain water. After letting it sit for a minute to drain off, I dry the egg, then take it into the house, and put it into the egg carton with all the other eggs in the fridge.
I just can't eat duck eggs when cooked "sunny side up," too slimy. But scrambled is fine, especially if I am adding things like bits of onion or bacon. And using them for baking is wonderful, but I don't do a lot of baking.
I don't know how long they will keep in the fridge, but it is at least weeks. So far, the only negative I have seen is the occasional piece of "meat" inside the egg, that I don't see until the egg is cracked open. But this hasn't happened very often.