The crop is located on the hen's right on her chest. At night it should feel full and rounded, about the size of a tennis ball, depending on the size of the chicken. In the morning it should feel flat and empty, as the food will have moved down into the gut and have been digested.
What do you feed, including treats?
I don't think she's dying. Can you feel her crop first thing in the morning before she has anything to eat or drink and report whether it is empty or not? And if not, how does it feel? Hard, soft, doughy, gritty, etc.? Let us know and we'll go from there.
Do you mean her comb, the fleshy thing on top of her head? The wattles are under her chin, behind the beak.
As for the feather breakage on her neck - do you have a rooster?