No
You will have to contact your local police. NY has a patchwork or laws.
If you are talking about a permit to drive, you have to be 16 years of age. (Hopefully they will make it higher with all the teenagers causing accidents now)
There is no learner's permit in NH. After reaching the age of 15 years and 6 months one may drive with a legal adult.
A NY pistol permit doesn't serve as a carry permit. As far as carry permits go, New York is a "may issue" state, which pretty much translates to, "we don't issue, unless you have some serious connections and can throw lots of money at us".
21, which, incidentally, is the minimum age to own a handgun under federal law. although if you are active duty military and / or have an honorable discharge from the armed services the age limit is waved.
No you don't need a permit
Federal law says you must be 21 for a handgun and 18 for a rifle.
No. Your NY learner's is not valid to upgrade to a Texas license. You would have to get a Texas learner's permit.
As long as you are conforming to the limitations and restrctions of your NY learner's permit, you can legally drive in any state.
No. You'll have to start the permit process over again.
Not without a person over the age of 21 in the car with you (:
Since you can post on the internet, do a search on NY gun laws.