Eggs are good for about a month. To tell if an egg is good do a float test. Put the egg in a bowl of water. If the egg sinks it's good and can be used. If the egg floats it is old and should be discarded. Cleaning and storage are a matter of personal preference. Some folks wash eggs thoroughly and store in the fridge. Some folks just wipe off any feces and store on the counter. Neither is right or wrong- just a matter of personal preference. Eggs stored in the fridge last longer than ones left at room temperature. Eggs that are unwashed last longer than those that are washed. Eggs should be collected each day to prevent things like egg-predation/accidental damage, but some folks collect eggs several times a day. Again, it is whatever works for you.
Good luck.