Greetings 
Zerex,
Your bird is definitely ill. Honestly, there are numerous illnesses that can cause lameness, discoloration of skin, neck weakness or twisting, etc. Even a vet would have to start with a CBC, and x-ray. If you could take her to see a vet, that would be the very best thing to do. If not this will not be easy to diagnose.
At this point, we can assume it is not frostbite. Frostbite causes necrosis of the flesh starting on the outer most areas of exposed flesh, the comb, wattles and the toes. Necrosis is purple, black dying flesh. She could have a slight freeze burn on her comb, which will be white tinged. But that would not explain her paralysis.
A discolored leg is a result of poor blood supply. The major blood vessel in the leg could have been injure after jumping off a high perch. It can also be injured during a break in the bone. But the fact that she does stand on that leg is most of the videos does not support a broken bone. Certain disorders can affect the blood flow to the leg. She could have been born with a vascular problem. I had a pullet like that. She died at 10 months old.
You say she is eating, drinking and pooping OK? If that is the case, those are good signs.
So, here is what should be done for your sick chicken.
- The sick chicken must be in a hospital crate, with mild warmth of 70 degrees. A heat lamp with a temperature gauge will allow you to monitor the heat.
 
- A fecal float test should be done immediately. Parasites can be a cause and contributor to poor health in a chicken. And often, interfere with recovery of a curable illness.
 
- She should have food and water available during the day.
 
- Administer, a good multivitamin supplement twice daily, for two weeks, to boost her immune system. Nutri-Drench is a good multivitamin that many chicken keepers keep handy. Pet-tinic is another good multivitamin, it contains iron, which is essential for treating injuries.
 
- Good nursing of the hen will help her on the road to recovery. This not only entails feeding, but also hygiene of the hen. Keep the vent clean of feces. A chicken diaper will save you a lot of work. A diaper eliminates the need for messy wood shavings or straw in the crate. But you will need to empty the diaper at least twice a day.
 
- A chicken with extreme weakness and/or paralysis does better in a small soft doggie bed cushion (use a diaper, or cover with a puppy pad or paper towels). Rolled towels placed around the chicken, can also be used as bolsters to steady the chicken while sitting.
 
- A sling/wheelchair is helpful. The chicken can be placed in the sling/wheelchair for a few hours each day to improve blood flow and mobility. 
 
Simple tub/box sling
View attachment 2543411
This sounds like a lot of work but it is part of caring for chickens. Once you have a regimen set up it will go smoothly. All animals get sick at one time or another. If we don't have a vet, we keepers need to educate ourselves in the proper care of the sick animal. Of course there are some things we cannot do like a trained vet. For instance, x-rays, and other tests are beyond our ability. So then, we must do a good examination of the chicken, and list what we see. Look in the chickens mouth and throat for lesions, lumps, growths. Take notice of any unusual or foul odors in the mouth or body. The condition of the poop will be one of the most helpful diagnostic tools for the keeper. What is the texture, Is it watery, bulky, or greasy? What is the color, is it grayish green, neon green, yellow, brown, white, black? What about the urates (the white part on the poop). What color is it, white, cream, light yellow or dark yellow. Also note whether the chicken is releasing a lot of urine when pooping. Yes, chickens release urine. 
These are my thoughts on your hens illness. I hope I have been helpful.
God Bless you and your flock.