Why does my cat bite my hair?
It's normal for any animal to bite the hair. Cats tend to do it as a hunting strategy, as a way to help get a good grip on their prey, and to help identify it.
It's not a good idea to keep a cat indoors all the time, but it's a good idea to keep your cat out as much as possible. You'll learn when it's appropriate to take your cat indoors and when it's not.
Kittens and older cats are also at risk of head lice, a serious problem in the wild but not too common in indoor cats. Many cats' hair will naturally get matted and thickened after they reach adulthood. When they go to brush it out, they are not going to want to get into a battle with their own hair, so if your cat is matted, gently brush them out before you take them outdoors.
A lot of people like to "re-dye" their cats hair to different colors. This is a terrible idea. Don't do it. Your cat's hair is their only true identity and they want to keep it that way. If you must take your cat outdoors to get their hair clipped or dyed, only clip or dye them if it's only one color. Even then, use natural dye on a very short cut, and then just trim it out after it's dried.
Your cat isn't biting your hair, she's just scratching. It's a normal instinctive response to an object you find uncomfortable to touch (like fur). In most cases you won't notice her doing it and it's a non-issue. If you're concerned, let the cat scratch in private without making a fuss. Once she feels more comfortable with the scratching, you can allow her to let her claws loose.
I had a very similar problem with my cat when he was young and very energetic. Sometimes he would bite me or bite some other unsuspecting (or unaware) object. I'd just get down on the floor and let him go at it for a bit. At the same time I'd get a towel, towel it down and just let him do his thing. Eventually, his teeth would become more accustomed to other objects and he stopped biting.
Cats instinctively scratch things in their mouths when they are not comfortable with them. A cat needs to have a firm, unyielding surface to bite. If the cat bites you or your clothing, it is a sign that the cat does not have a firm, unyielding surface on which to bite.
As a rule I never have let my cat or dogs scratch on my clothes (excepting their nails). But for many this can't be done (not to a dog and most of us won't tolerate it for a cat) and having lived in a house with a dog or having cats for a long time I know the behavior.
So what I'd suggest to try is to get a carpet mat. I haven't tried it for myself yet but I've read some where people found this worked wonders. My father was a vet and for long years we used to deal with sick cats, dogs, rabbits and even birds. What I can say from personal experience is the cat would be much more at ease on this carpet mat as long as they are allowed to go at it on there own. You can also put other mats around as you have more time.
Try getting some carpet tape, you can put a piece on each leg/paw before they've had a chance to dig up the carpet, and then tape it all over the rug once they've dug up the carpet and your at risk of scratching up your nice new rug.
This might be helpful:
I personally like the method that the vet used to teach my brother and I as kids - place large heavy objects in the corners of the room. When he did this his cats would come out of the closet and sit in a corner for an hour or two, and then they would return to the closet. Once they were done this would happen several times per day. This was a technique that he used for any type of behavior modification he was trying to accomplish, and it was very effective.
Another technique was used in this video by Mr. Pibb, which I have never heard of before, and it appears to be effective.
To quote one of the commenters on this answer, this is a very good question, but the behavior is a cat thing and not a dog thing. If your cat gets this behavior after being in a new place,