Congratulations!
It isn't always necessary to wash eggs, but if they have dirt or mud on them (which can often happen after a rain) you will need to wash them.
I haven't heard that people leave eggs outside that long. Usually people bring them in on a daily basis. However, you don't have to put them in your refrigerator right away. Lots of people just leave them on their counter for a while until they use them.
Your hens need to go broody in order for them to sit on eggs and hatch them. When a hen goes broody, she will stop laying, sit in the nesting box all day and try to hatch the eggs. Just do a search for "broody hens" to read more. And if you place your neighbor's eggs in the nesting boxes, your hens won't do anything to them.
Good luck with your chickens!