I grew up in NY, and at age 18 was granted a pistol permit. This was in 1975.
In those days, it was entirely up to the judge who approved the permit. And each judge was different. I knew of one case where a boy was approved at age 14. Apparently the judge in his county was ok with letting him have a pistol to manage the trapline he ran near the Canadian border.
The Judge in Saratoga county would only approve a permit if the stated reason was "Hunting & sporting purposes". He would deny it if it was for "self protection. The Judge in Albany county would only approve permits for self protection, and would deny permits for hunting & sporting use.
It was extrememly uncommon for anyone under 21 to be approved, but if the judge allowed it, you could be.
Thats the way it was back then. I left NY for good in 78. Sorry I don't know the current state of the law in NY.
All else being equal (and it almost never is) bigger bullets tend to work better.