When you rinse eggs, you remove the bloom. The bloom is a natural "top coat" that the hen applies to the egg at the very end of the laying process. It seals the egg to keep out dirt and germs. Eggs with the bloom intact last alot longer than eggs with the bloom removed.
To answer your first question - I don't wash my eggs. I keep my nestboxes as clean as possible and most eggs come in from the coop ready to go in the fridge. For the occasional dirty spot I use a paper towel to scrub the spot off. Really dirty eggs, for example ones that were dropped on the floor of the coop, get fed to the dogs.
I choose to refrgerate, others don't. Either way is fine, but eggs on the counter do lose their freshness faster I've read.
It takes a hen roughly 26 hours to lay an egg. The way that works in my coop is this - if my hens visit the nestbox at 10 tomorrow morning I can expect them to visit around 11 AM the next day. This hourly staggering continues until the day arrives that it's too late in the day to lay, then they either start over the next day or skip a day and start over early in the morning the next.
Yes, most of my hens cycle together, which is the same phenomenom seen in female humans living or working together. On any given day most of my hens visit the boxes at roughly the same time; with one or two waiting till just a bit later than their sisters.