When mine first started laying I used to leave one egg in the next box to encourage the others to lay. They don't seem to need it anymore, but if you concerned they need encouragement it couldn't hurt to leave one there, and if you are afraid they will break/eat them, you can leave a fake one in the nest. I don't think they will damage the eggs tho. If one is already broken they may eat from it but if yours are like mine, they are more concerned with laying and then getting back out in the yard to play. If you do leave one in there, perhaps a couple of hours is long enough.
You do not need to refrigerate unwashed eggs. As long as the bloom is intact, that is the coating that they are laid with which is part of the immune system of the egg, they do not need refrigeration. If you wash them, refrigerate them asap. I've kept unwashed eggs for as long as 3 weeks (that is the ones with the bloom on them) and refrigerated eggs for up to two months. I do not really know what the recommended time is but I'm sure you can find out online. Mine don't last long enough for me to be concerned about it.
Good luck with your hens! My girls are the love of my life and even though many people consider chickens 'livestock' to me they are like dearly beloveds. They come running when I call them and follow me around the property for walkies just like puppies. I had no idea when I got these little feathered bodies of joy that it would be one of the most satisfying experiences I've ever had. They just came into my life unexpectedly and I've been enraptured ever since.