When eggs are laid, they are covered with a coating that preserves them. So don't wash eggs when you collect them, wash them just before you use them.
To see if an egg is good before cracking it into the food you are preparing, either crack each one into a little bowl then dump the bowl into the food or the trash, or float each egg in water. If it sinks and lays on it's side, it's good. If the large end picks up again, it's not totally fresh, but should be fine. If it stands on it's end, it's probably fine, but you might want to use it for baking rather than for a breakfast egg. If it floats, it's bad.