I think you must use on with dates as a rule
- The exam will be on 3rd March or
- The exam will be on that date.
There's a simple and clear explanation of prepositions of time on this webpage:
Prepositions of Time, Place, and to Introduce Objects
I hope you find it useful
The form "will be having" is a future continuous, so you are supposed to use it to describe something that will be happening at a certain time in the future.
Ex: This time next week, I'll be flying to Paris.
Don't phone me at 9. I'll be having a bath.
Does that help?
Student or Learner