R.L.
I think you have a wonderful opportunity here. Our children sometimes choose children for their friends who we wouldn't necessarily choose, but remember, they are six! There is learning possible for both the girls, and for you. I recently attended a high school graduation party for a young man who was quite wild as a little guy. When he visited I always had to put away toys that I didn't want damaged, and we had some really interesting chats about how to ask for things, and some of the words coming out of his mouth. He's grown up to be a lovely young man, off to college soon, very polite and engaging, and still quite a mischievous twinkle in his eye.
At the age of 6 they are so open to learning. Don't let the friend put you on the defensive. She is just trying to understand the differences between her family and yours. You can patiently explain why you love your house, why you enjoy the shows you enjoy. Don't put her choices down, just explain that people feel differently about these sorts of things. Have a sense of humor, "I know, I'm just a silly old mom who prefers Dora to Lady Gaga." It also sounds like she might have some attentional issues, so I wouldn't take her calling other friends or having a hard time finishing an activity personally. Let her know what your expectations are, "please don't run in the house," "let's try to finish playing this game before we start another," "your mother is coming at 3 so we don't have to call her now, let's do this until she comes." I think if she sees you as being on her side, she is more likely to learn some things from you.
I think I'd also consider what your daughter is getting out of this friendship. Your daughter might be letting you know that she is interested in more grown up things, and while Lady Ga Ga might not be the answer, maybe there are other teenage girls who she would be interested in watching that you would be more comfortable with. Little girls are always attracted to pretending to be grown-up, but help them both see that "grown-up" has many varieties.
Sometimes shy children are attracted to more outgoing kids. If this girl is truly mean to your daughter, you can help your daughter think about whether she really wants to pursue this friendship, but in the meantime, she could be experiencing this girl as alot of fun, as someone who is more comfortable acting in the world and there are good things about that. The friend might also really like your daughter even if she doesn't show it in the way that you think she should. Your daughter's sweetness and calm could be appealing to the friend. She might even be wanting the sort of limits there are in your home, even if she questions them. Girls like that are also sometimes good friends to have in order to meet other friends, especially if she reaches out to other children when they are together.