B - you've got 2 challenges going into this one - 1) she's 13, 2) she has 2 sets of parents with apparently different sets of rules, yours being the stricter. My opinion is that I think your rules about the music and the cursing are totally appropriate. I was 30 before I could comfortably say "damn" in front of my parents. The music they weren't too worried about -- if anything my brother would crank his Ozzy Ozbourne & AC/DC and then lock his door so EVERYONE had to listen to it. I still dont' think even most heavy metal is nearly as rancid as today's rap tho. The approach my parents took with swearing (and note I was a really good kid, but it was acceptable among my friends to say f***, etc.) was to let me know that 1) it was disrespectful and 2) it reflected poorly on me, my education, my command of the English language and just generally poor "breeding" and would count against me when I had to go out & make my way in society.
When you are dealing w/ a child (and 13 is still a child), it never seems to get you much when you just say "This is the way this is in my house and you will respect me'". Because the kid is now "13" and starting to feel more worldly wise, etc. so if you can expand the purpose of not swearing and an awareness of the same disrespectful and ignorant use of the language in the music she listens to will continue to be seen as reflection of your daughter as she goes out into the world. And in that arena, swearing and poor choice in music are viewed as characteristics of lower, more ignorant, less respectful and respectable levels of society. Does she view herself as the trash of society? Then why should she act like it? If she acts like it, she will be treated like it.
She may not acknowledge this approach immediately but I think it might strike a chord with her about how she presents herself outside the tiny little world of her and her little urban wannabe hoodlums.... Just like you said, and I know every adult I know now, including myself, didn't appreciate the strict rules their parents gave them when they were kids because we all thought we knew it all and our parents were just old fogey sticks-in-the-mud who cramped our lifestyle and "just didn't get it". I think you need to continue maintaining your rules because without them she will lose her bounds of what is acceptable and what isn't as she grows up. She may fight you the whole way, but at least she will be aware of the unacceptability of her actions. Some people just grow up simply not knowing the difference and spend the rest of their lives trying to figure out why people don't accept them as intelligent and respectable people worthy of respect.
I hope anyway.
I also wanted to say that maybe when she acts like she doesn't need you anymore, maybe you can indicate to her that YOU need her. She needs to know that you will always be there unconditionally. Especially when she gets a couple years older and realizes that the big world out there is a bit tougher than she thought she was. Its when kids feel they have no where to turn that they are truly lost.