I got taken away from my mom when I was 15, and she handed her custody over to my dad. Now that I'm 17, could I live with her again, even though I'm in state care still?

You have to wait until you are 18 unless mom goes back to court and can convince a court to restore custody. However, if a court took you away only about 2 years ago, I doubt if enough has changed to change the custody order.

Your mom will have to file for custody of you and seek the state to turn you over. Most likely this would take longer ( if they even would) than your 18th birthday