No, you do not have to report it. Once you file your case, any income you earn and receive after that belongs to you and does not put into the bankruptcy case or go to your creditors. Generally, the only exception would be if you receive an inheritance within 6 months after filing your bankruptcy case.
No. You do t have to report it. It is post petition income and not part of the estate.
No. you don't need to report the post-petition severance payment. The notice you received was probably just a pro-forma finding that the chapter 7 case is a "no asset" case; i.e.: nothing for the trustee to administer.