This is normal, my cat was like that as well, she is siamese, this
breed is very intelligent, also, very energetic, don't worry, she will calm
down after she hits 1 year old. Now, just keep her busy and she will be tired, I bought her a feather cat teaser and she loves jumping up and down for it.
As for the biting issue, just loudly say Ouch!!
and she should stop, and if that don't work, do not beat her, press her head down to the floor for a few seconds and say firmly NO. It worked for my cat,