In New York can a 16-year-old be emancipated if living with their parents is causing them to be very depressed?

You can seek help, but it would be almost impossible at 16 -- the height of homronal imbalance -- to prove your parents are depressing you. To be emancipated, you need to show the court that there is sufficient need, that you can support yourself financially without any outside assistance and you have your own place to live, apart from friends and other family.