Good point on the rooster Ridgerunner.
Crack them in a separate bowl, and if any look/smell funny, feed them to the dog if you have one. (or back to the chickens, cooked of course) My dog gets any questionable or the occasional egg with a bunch of poop on them. I don't normally get poopy...
Yes, lock them in the coop/run for a few days.
As for the eggs, they should be fine this time of year. I'd crack them into a separate bowl one at a time and check them out. You can do the float test on them. Put water in a glass or bowl and add an egg. A fresh egg should sink to the...