Is it also safe to assume check-in/bag drop counters are open for BA after the morning flight has departed but well before evening flights? I ask because I've never tried, and would wonder whether/why they would keep check-in staffed for many hours during which there are no departing flights.

Another suggestion if the OP is at all interested would be to consider heading east instead of west for a few hours of repose. Assuming OP has no desire to drive oneself around, booking a car with driver for the day, although expensive, might be a lot more pleasant. Jones Beach is about 30 minutes from JFK, Robert Moses around 50 minutes... hourly booking for a service like Blacklane is a touch over US$600 for an eight hour period, but they can pick you up at your cruise terminal, take you wherever you want, wait with your bags securely locked in their car while you have lunch or a stroll on the beach or whatever, and keep an eye on time and traffic to make sure you get back to JFK in time to fly. If the budget permits, this is certainly what I would choose to do.