Welcome to BYC! Please make yourself at home and we are here to help.
Congrats on your new eggs!
There is no need to wash eggs unless you want to. Water washes off the unique eggshell coating called the "bloom" which keeps bacteria out. So, if you do wash the egg, either rub it with vegetable oil (which acts as a different covering) or eat the egg sooner than later. (Here's a link on cleaning eggs explained further.
https://www.backyardchickens.com/a/cleaning-and-storing-fresh-eggs)
If eggs happen to have a little poo or bedding on them, simply brush them off or use a dry paper towel to remove it.
As for determining if they are good or bad, there are several ways to tell this. If you are wondering if they are fresh, place them in a bowl of water. If they float, they are old and if they sink they are fresh. Also, cracking the egg open and looking at the yolk can help determine freshness. If they yolk is dull, runny, flat and yucky looking, I wouldn't recommend eating it. (Give it to the chickens or the dog instead.)
Hope this helps. Good luck!