Yes, for starters.
To register to vote in New York State you must be a U.S. Citizen and 18 years old by the end of this year.
Can you please explain to me if is there any link between this proof I have to send and my discharge date?
Well, yeah. If your discharge happened before January 1, 1950, then you should submit a Honorable Discharge Form WD AGO 53, or WD AGO 55, or WD AGO 53-55, or just an "Enlisted Separation Paper"