If they’re dirty, you can either sand off the soil with fairly fine sandpaper and/or wash in warm water. No need to buy the fancy egg wash products. As has been said, wash just before using them.
You *can* leave them unrefrigerated, but I don’t. It seems to me they stay fresher in the fridge. As they age, the white thins out. Really fresh eggs don’t spread in the pan as much as older ones do.
Also they can be a real challenge to peel if you boil them. I steam mine in my instant pot for 6 minutes, quick release, then unload them directly into ice & water. Peel as soon as they’re cooled off and store in the fridge.