If you are working currently in the US on a H1-B visa, you can start a business, as long as you do not work for that company. You can only work for the company that sponsored your H1B and only the kind of work specified in the petition. There are no exceptions to this rule.
However, you can establish a company and appoint a director or CEO to run your business. You would essentially be an absentee business owner and would be restricted from having any significant role in the company, which is a deal breaker for most people.
Yes you can. However you might need another director in India. This will depend on the type of company. You need to get hold of a good CA.