No, you contact the New York state department oif health, the instructions for obtaining a duplicate official birth certificate are on their website.
Possibly, but I would try researching their website to find out, or calling them during normal business hours. I don't know of any local government that would issue a birth certificate through the internet. To be issued a new birth certificate, I would think you would need to physically go there with some form of government-issued identification (driver's license, state ID, passport, and/or social security card, etc), in order to prove who you are.
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