Once you are over 18, you become legally responsible for providing consent for all medical treatments (provided that someone does not have legal guardianship over you). You have the right to implement a living will. This means that they have to respect your wishes (or face the legal consequences).
This is easy if the circumstance is minor and you are conscious and cognitive. However, the situation may become very complicated. For instance, if you were unconscious and there were no way to alert the medical personnel that you have such a card or a will, then they will take action that is "in your best interest" and then figure all the legal aspects later. Or if a situation is not clearly addressed in your living will, they will try to contact the next of kin for guidance. Also, if your parents are paying your bill, they will be notified with the statement.
If you are conscious and/or they can find these documents in time. If they ignore them, then you have legal remedies to take (of course, your parents would have already been notified). However, in a true emergency, they may not have the documents on hand; and they must treat you to the best they can (which may include contacting your next of kin).
They can't without facing legal ramifications. Also, the card must clearly state so and have your signature, etc. on it.
They can do what they want; but if they have the card and will and ignore them, then you have legal remedies against them.
I don't know if the contact card has any legal "teeth" to it. You can't just write it down somewhere. You are best protected if there is a legal document with your name notarized, etc. Also keep in mind that if there is danger to others or to yourself, then medical personnel are required to report.