Should I be friends with her?

Heck no! She threatening your family and just making you look bad on Facebook! I've seen girls like this at my school, they just pretend nothing happens and when they are bored sitting in the dentist office they decide to become friends with you and again and start texting you again. If she's old enough to have a facebook, she's old enough not to type crap that her mother told her to type. And if she gets that influenced by her mom to type all that mean stuff, then don't talk to her.