Dear John,
There are some sites specific to New York City. One great one is
https://italiangen.org which has databases for births as well as marriages by groom’s or bride’s names. Free. I cannot remember if there are deaths there.
The municipal archives in New York state have vital records of all kinds. You may have difficulty determining which municipality has jurisdiction, and therefor the records. Obtaining a copy costs money, usually about $30 US. Some places can send you a copy by email, but some still mail a physical copy.
Don’t know about Central Islip Mental Hospital. You might get more response to your questions if you post on the G2G section of WikiTree. Maybe with a New York label.