Police and Jurisdiction

It is my understanding that a New Jersey Cop has no jurisdiction in NY. This meaning that an NJ cop is (kind of) not a cop in NY?

Now an NYPD officer.... does he have jurisdiction just in the city? Or does he have jurisdiction in all of NYS? (figure Mt. Vernon as an example)

Is an off duty police officer allowed to carry a weapon inside a bar and consume alcohol? (anywhere in NYS)

Is there a good website I can go to that will help me to answer these questions and (many more) about what police can and can't do?

I ask because sometimes I run into some pretty power tripy cops around here and I've been told that they tend to back off if you know the law.

I've had my car illegaly searched many times.... and they've found nothing, I'm not a criminal. (no plain site evidence, no smelling like alcohol or pot, etc.)

I know a little bit about the law and such.... but I've always been too scared to deny consent to an illegal search out of fear of being arrested. And isn't it their word against yours anyway?

I don't hate cops.... just the handful of rotten ones that spoil it for everyone else.