Some stop, and some don't. The molt is hard on them, so it's nicer for a hen if she stops laying during the molt and dedicates her energy and protein intake to her new feathers. You can help them out, especially since they are laying through the molt, by upping their protein a little. Give a few meat or fish scraps, some mealworms, bits of hardcooked eggs, or some higher protein feed during this time. Also, black oil sunflower seeds will benefit their new feathers.
Yes, give them the whole thing, shell & all. The shells have something in them that's good for feather conditioning, I can't remember what it is. I guess a search of BYC might reveal it, tho. I think I read it here.