I am a New Yorker, just moved to San Francisco and made the cross country flight with my 2 yr. old several times. It's not easy with two parents, let alone one. I always bring the car seat, umbrella stroller and a large tote bag on board; these are sort of the essentials. I bring the car seat because I get my daughter her own seat on the plane and she sits in it but if you are holding your child you could check it. I think you will most certainly need it on the other end for the taxi/car ride from the airport if not anything else. And if you take a taxi or car in NYC you will need it, yes.
If you can bear the expense it actually is much better having them sit in it in their own seat on the plane as they are strapped in and don't wander or fuss and have a chance at sleeping that way then if they are on your lap. Plus they keep their hands and feet to themselves rather than disturbing another passenger the whole way.
For the plane I stuff as much as I can into that tote bag; I put diapers, wipes and cream in a zip lock freezer bag that I can pull out to go to the restroom as it is too tight to bring a bulky diaper bag in. I also pack a small cooler bag or lunch sack with ice pack full of snacks; juice, yogurt, lollipops (sucking helps their ears) pretzels, fruit etc. because the food on the plane is never suitable for kids and they don't always have whole milk etc. My daughter is a reader so I bring allot of books; soft cover only to cut down on the weight and some coloring books or a travel size doodle pro. I usually buy a half dozen small toys I can pull out at intervals to keep her entertained. Nothing with pieces to get lost or noises which may disturb others though.
I also make sure to bring a change of clothes for her and an extra shirt for me as we have had some mishaps now and then and 6 hrs. is a long time to sit in something wet or soiled. A favorite blankie or stuffed animal also helps.
As for staying with friends or family you might want to ask if they could borrow a crib or pack and play for you to use while you are there; I've been surprised to find that most of my relatives and friends know someone with an extra or one they have outgrown. I've borrowed highchairs too when my daughter was smaller.
Good luck!