There is not a whole lot I can add to the excellent answer Martissimo just gave you. Your cat sounds completely normal! Kittens are rambunctions little critters!

Your kitten may perfer a horizontal rather than vertical surface. See if you can find one of those cardboard scratching pads and offer it to her. She may or may not use it. The carpet where she has scratched can also be covered with aluminum foil for a while until she is trained to a post or other more preferable scratching surface.

Using a featured kitty tease has been recommended as a means of lowering aggressiveness in kitties. It might be a help with the biting. Many cats do bite when they've had "enough" petting. You are right to stop as soon as she stops purring. Also watch her ears. When they turn backward that is another sign your petting session has been declared over.

Good luck and enjoy your new baby. Before long she will be a sleepy mellow kitty and you'll wonder where that kitten went.
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Smile, it will make them wonder what you are up to!

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