Okay, first of all, Happy Mother's Day.
Now, my kids are 17 and 13. I learned a valuable lesson a long time ago from my aunt when my cousin went through some very interesting phases. It is only hair. It will grow back. My cousin went through a whole mess of phases between 12 and 18. (BTW, she is now in her 20's, married, working, and fine.) Anyway, she had hair that was pink, purple, black, mohawk, half a mohawk, etc., she had a mohawk at my son's christening. Anyway, she was just trying to find her own self. When everyone was giving my aunt a hard time about her daughter, she just said, "It's only hair." Now mind you, my aunt graduated first in her class at MIT and was valedictorian. She is one smart and classy and well groomed woman. So, this advice hit me really hard.
Anyway, my daughter is 17 and she has gone through red streaks, blue streaks, fake nails, etc. She doesn't have piercings, not even ears, since she is afraid of needles so I am not worried about extra silver peeking at me or tattoos, but she is just expressing herself.
You have to go with your instincts but I'd compromise. Ask a hair salon about the best way to go about the hair thing. Talk to her about how her hair will be discolored after the temporary rinse and try the hair spray. She may decide that she doesn't like having black hair. Make her learn about taking care of her hair, etc. My daughter got away with the streaks and hair colors because I made her buy them herself. I just signed the permission slip at the salon (and the blue streaks looked really good on my daughter - they matched her eyes). You daughter needs to learn to take responsibility and learn the consequences of dying hair too often or the wrong color.
This could be a great opportunity for you to start forging an open relationship with your daughter before she gets into her teen years. 5th Grade is when they start making the transition to middle school where everything changes and if you don't have open lines of communication now, you will never get them because next year, her friends will be more important and smarter than you (to her).
As for the nails, if she has an allowance, tell her she would have to pay for them herself and maintain them herself. When she gets a load of how expensive it is to maintain, she may change her mind. For ages now, my daughter, my mom, and I hit the nail salon once a month to get manis and pedis together. It is two and a half hours of uninterrupted girl time. We laugh and chat while we get our feet massaged. I only pay for the pedicure, she pays for everything else.
Yeah, she might be a bit young to you, but in the blink of an eye she is going to be grown and you will wonder what happened. Forge that friendship and trust now and you will have a great relationship with her as a teen. People don't believe me, but my daughter tells me EVERYTHING. By not judging and pushing my views on her, she has made some very intelligent decisions and when she has made a mistake, she comes to me and we talk about what happened and how she feels would be the best way to fix it.
As for the play? Well, go with the hair spray and then talk to her about the expense of dealing with nails, etc. Sit down with her, make it girl time instead of a fight. Go to the beauty supply store and read all the products together. Talk to her calmly about how each product may effect her beautiful hair. Ask her is she wants to take that risk. Remember it is only hair, and it will grow back but having her laugh and talk to you and spending the time together is so much more lasting.
However, don't go against your feelings - you are the mom and there is a food chain - LOL. State how you feel about things. If she wants to change your mind, you are teaching her negotiation tactics and debate skills. ;) Don't not say no to be her friend. Stay her mom but open yourself up to the possibility of also being her friend. Who knows, you may end up like me with all the girls wanting to hang out with you because you are honest with your information. When I tell one of my daughter's friends that the color choice they are looking at is nasty, they believe me. And, when I went to color my hair to cover that nasty grey coming in (wink), they picked my color and it was perfect.