The next morning would be correct.

I think "On the next morning"

If you have to use a preposition, it would need to be 'on'. If you are narrating a sequence of events, it's possible to say, 'On the next morning,...' as an alternative to the simpler and more common 'The next morning,...'. You cannot use 'In' here.

It's neither. We can simply say "the next morning." You can also say "during the next morning" or "throughout the next morning" if the event you're describing took place for much of the time period.

We don't really use either of those responses, we would just say 'tomorrow morning.'