Oh, you poor dear. Sounds like you're having a tough time.
I think by piercing, you mean that you want your cartilage pierced, right? Up high on your ear?
I don't think I would have let my daughter do this when she was this age, although she did get it done last year when she was 15. I have mine pierced and couldn't really find a reason why she couldn't.
Perhaps you could ask your mom for a reason? I don't know how mature you are, but that sometimes plays into the "no" answer.
As far as dyeing your hair, again, you should ask your mom why she is saying no. Perhaps it's a money factor. Once you start getting it colored, it costs a lot to manage it. You have to get the roots done every few months. This can really add up. And with this, I have to say that I don't think I would have let mine get her's done when she was that age.
I think that there are some things you should just be a bit older for.
But, that's just my perspective.
What you should do is be respectful of her decision, be obedient to it, and try to have a heart-to-heart. Maybe ask her when she might allow you to do those things so you have something to look forward to. Say, by your 14th or 15th birthday. Something like that.
Hang in there. It's tough to be a teen, but it's just as tough to be the parent of a teen.
Been there and doing that.