I dyed my hair blue at that age. 
I guess the real issue is whether the 'rules' have been issued to you in advance, or not. If they have, I'm afraid I think you will have to abide by them. If not, I think you should fight it.
I was "told off" by my wanker of a head teacher for having blue hair and he threatened to send me home, until I pointed out there was precisely nothing in the school rules that said I couldn't have blue hair (had checked beforehand!) He had to back down after that.
As for the wrong impression etc I agree that it is daft. I got good exam results all the way through and unlike many of my peers I never got arrested whilst wearing school uniform.
However, if the rules are there, and have been issued to you all, I think your daughter will have to compromise. It's not up to you to decide which school rules do and don't apply to your child. It's up to her to find creative ways of flouting the spirit of them whilst ostensibly toeing the line
one of the great joys of being a teenager.
Ultimately if you decide to breach the hair rule, X's parents can decide to breach the sick note rules, Y's parents can say it's fine for their little darling to give someone else a smack because they're stressed... Chaos ensues.