I store eggs on the counter in an open carton using the 'cups' for salable eggs and the cover for my eating eggs, unless they have been washed.
They get eaten or sold way before they'd 'go bad' and it's just easier not to deal with refrigerating them. The temp there is moderate all yer so no worries about overheating. I rarely have 'too many' eggs, I have 'extra' customers for times of plenty, but would refrigerate unwashed in cartons if I did and I've read that using an airtight container will deter moisture loss.
I don't wash eggs unless they are very dirty, then will refrigerate or use immediately because thorough washing will remove all the protective bloom(cuticle).
Eggs should be washed in 'water warmer than the egg'.
Simple physics, using colder water will cause the egg contents to contract, causing any 'germs' on exterior surface of egg shell to be pulled into the interior of egg thru the shell pores. Using warmer water will do the opposite, makes perfect sense to me.
I don't use any soap or other cleaning/sanitizing agent, just rotate in my hands to 'scrub' all surfaces area of egg shell. Then I air and towel dry before placing in the fridge.
If you are washing eggs for sale to the general public, other requirements may apply, so check your state regulations.
The 'float test' will only show that egg has lost moisture and may be older, will not tell you if it is 'bad'. Unless an egg is cracked and oozing, or stinks, the only way to know if eggs are 'good' or not is to break them one at a time in a separate dish and use your eyes and nose to assess before putting in pan or recipe.