I am answering some questions on my known knowledge
1. Gratuity is not about death or alive; if the employee completes his 5 year service with employer, gratuity can be come, if the employee dies, the nominee can get the gratuity, so in your case check whether the employee serve 5 years or more .
2. Insurance- if your company is tigh up with private insurance, see the insurance manual and the same can employee family can claim; if not ESI, if employee has ESI the employee family can claim the death benifits from ESI, on both cases employer is 3rd party for signatures and approvals.
3. some companies can mention health and safety line in offer letter that means if the employee dies with in the office premises by the mistake of a machine or employer accidentally then that clause can be applicable, on your case i don't think those two are points.
4th and 5th questions; i have no clue of what you are asking about.
but as of i understood, if the employee death belongs to employer you have a right to claim the death or file a case towards employer, if the post mortem report said it is natural death, no one can do anything, if the employer has a death fund in his company, they can help you on some money.