I am pretty sure you can drop your bags off and go off exploring and then check in and board the ship. I’m not so sure you can check in and then leave again.
They probably are not adding or subtracting items, but they may be rearranging the days they will offer each menu. The app is simply unreliable, especially this far out from your trip
With any time dining you can go without a reservation and yes you may have a wait if you go at a popular time, but they will give you a buzzer and you can wait at a nearby bar or lounge
Sometimes, but like the state resident discount they come and go. You can ask but don’t be surprised if the cruise you are inquiring about does not have it