i can't imagine telling a nearly 17 year old she couldn't dye her hair. i started dying mine all sorts of colours when i was about 13. i can sort of understand not wanting her to get tattooed, but piercings are another thing i can't really imagine telling my older teenage daughter she wasn't "allowed" to do. i was 17 when i got my first tattoo and it is a truly beautiful piece of work. i had a fair number of piercings by then too, and the great thing about piercings is that they can almost always be removed without any issue... hence the only one still remaining is my nose piercing i got in my 20s.
look, i get that it's not something you think is a good idea, but i honestly don't think it's your call. body mods (and hair dye does NOT fall into that category... that's along the lines of make-up in my book) are something that truly only hurt the person who does them. she's not going to be unemployed for life because she has a few tattoos... i'm an elementary teacher and no one has ever said a thing to me about them (nor has it affected my ability to get a job, even in very conservative atmospheres). a lot of people 30 and younger, maybe even the majority, have some kind of minor body mod at this point... it's just not going to be something that affects her life with any severity in the future. as long as she is paying for it herself, going to reputable shops and is aware that her choice of placement, subject matter and size of body mod are in some way permanent, it's all good. maybe she WILL regret her first choice of tattoo, or get bored of her conch piercing, or have someone react negatively to her hair colour. these are learning experiences though, and nothing that is going to seriously ruin her life.
just so you know where i'm coming from: my parents are conservative christian missionaries. when we moved back to north america, i was really drawn to alternative culture and within a couple of years was getting pierced, tattooed etc. i don't agree with all the decisions my parents made, but one thing i really appreciate was that they emphasized caring about my ACTIONS, not my appearance. my mom used to help me dye my hair purple, she said she thought my tattoo was pretty (wasn't thrilled i got it though), she never criticized my choice of piercing, and she never banned me from getting them. i think it's a pretty normal part of teenagerhood to experiment with one's appearance, and the resounding message i got from my parents was that they loved me no matter how freaky or ridiculous i looked to them. the message you're sending to your daughter right now is that you think she will look terrible with blue hair, and you want her to 'look normal' ... basically, that your approval of her is tied to her appearance.
finally, this will either become a pretty permanent part of her life, or she'll get over it and move on. i haven't gotten a new tattoo in a loooong time, haven't dyed my hair bright colours since university and only have my nose piercing now... i'm pretty clean cut looking now and no one would probably guess how i looked at about 17-18. making it into a battle seems silly and counter productive to me.