From my experience, subway from Manhattan to JFK might be actually faster, let alone cheaper.
Yellow cabs have fixed fare from anywhere in NYC to any city airport. This is written on each car. This is why, probably, they tend to refuse: this is not that profitable.
Jumping in is best practice but not always possible. Cunning drivers would sometimes ask you where are you going before opening doors, though legally they don't have to, and the passenger should tell them once in the car.
/* This is another example how regulations aimed to make a service affordable result in its becoming inferior, yadda yadda*/
Subway from manhattan is almost certainly not faster unless there is very heavy traffic (which of course is fairly common on weekdays). If you want to take transit, LIRR from Penn Station + Airtrain is generally the way to go, even if you have to take the subway to get there (except maybe from downtown, where its a wash)
Yellow cabs have fixed fare from anywhere in NYC to any city airport. This is written on each car. This is why, probably, they tend to refuse: this is not that profitable.
Jumping in is best practice but not always possible. Cunning drivers would sometimes ask you where are you going before opening doors, though legally they don't have to, and the passenger should tell them once in the car.
/* This is another example how regulations aimed to make a service affordable result in its becoming inferior, yadda yadda*/