Is it possible for a hen to lay two eggs in one day? Absolutely. If a hen releases two yolks together you can get a double yolked egg. If she releases two yolks separated a bit in time she can lay two eggs. That happens more often with a pullet just learning to lay or an older hen nearing the end of her laying life but even a healthy 1-year-old can have a glitch and release two yolks.
Often when that happens the eggs are not both perfect though. Often does not mean always so bear that in mind. A hen often only makes enough shell material to cover one egg each day. The second egg is often thin-shelled or no-shelled. Same thing with the pigment that coats your eggs brown. Sometimes that second egg is lighter in color. Not with yours. Often, if they lay two in one day they skip laying the following day. While they are in the shell gland getting the shell put on they are right next to each other. If two eggs are in the shell gland at the same time you get a flat-sided egg. It's a deformation since they were touching.
You will notice that I use weasel words like "often" or "sometime" a lot in this. That's because often does not mean always.
Yes, but not that change. The brown goes on at the very end of that egg being in the shell gland. When you crack those eggs the inside and thickness will be white with the brown only in the outside of the shell. If something causes a hen to lay her egg early she may not get all of the brown on it so it can look really different, maybe even white. If something happens to cause the hen to delay laying the egg some extra pigment may get deposited, but not enough from your other hens to give you that deep chocolate brown. They don't have that kind of pigment.
It is possible she laid two perfect eggs in one day. Since you found the extra egg when you were cleaning out the nests is it possible one was just hiding in there? To me, this makes the most sense