They will check your documents to make sure you have the proper paperwork for where you are visiting. If you for example need a visa and don't have one, they will not let you board.

That will happen when you check in.

As to the hour, if you are ontime, if you can check in to both flights, if you don't have to reclear security at jfk because of a terminal change and all that you will be OK, but even if everything goes 100% right, it's still cutting it tight.

If there's an option for more time I would take that.