Basically, you have to collect your checked luggage, clear through entry formalities in the UK and arrange your own way of getting to Gatwick. Many recommend the National Express coach service as it is door to door (with the train you have to change including 2 London stations). When you get to Gatwick you need to check your bag in and go through security / departures in the normal way.

Yes if your inbound is late & you misconnect you are protected, and at Heathrow there is a good chance they’d put you on a JFK flight from there.