I think Southernbelle answered your real question. I'll expand a bit.
The best sign of a broody is that she stays on the nest almost all the time , day and night. She normally will get up once a day to eat, drink, and poo, then hop back on the nest. Since she is holding her poo, when she does let it go, it is a huge really stinky mess. When you first see and smell it, you will think one of them is sick.
If the broody is on the nest, she will normally be very upset with you if you get near. She'll fluff up, growl, and possibly try to peck you if you get close or reach under her. Not all act this way so it makes it a little more difficult to describe, but most do.
When they are off the nest, many make a constant clucking sound. When they are on the nest, they often look like they are in a trance. Once you know what it is, you can recognize it.
A hen will normally lay an egg a day before she goes broody, then quit laying to set on the nest.
Sometimes a broody will pluck her breast feathers to insulate the nest and to sort of bare the skin that will be on the eggs. She can keep them warmer this way.
Probably more than you wanted to know.
As for your hen with feathers missing, you might check her for lice or mites if you haven't already. It's best to check at night as you will not find the red mite on a chicken during the day. That particular mite hides in cracks and crevasses during the day and attacks the chicken at night. I'll include a link.
http://ohioline.osu.edu/vme-fact/0018.html