For (2), you won’t have to collect it unless you specifically ask for it. Otherwise it will remain in the system and be automatically rerouted to whichever is your next flight.

(4) no need to make specific plans, most connections work well. The horror stories are mainly from the post-covid recovery period and have largely gone away. If anything happens on the day, head for the flight connections centre in either terminal 5 or 3, where there are plenty of BA staff to rebook you if necessary - as above, they can use other carriers if they need to. If anything looks dodgy/delayed before you leave NY, they can also sort you out there.