Bank Account

Not necessarily. Even though most big banks require that the owner be physically here, certain mini-tier banks will allow you to open a bank account online.

NOTE: Entry India has a relationship with a few banks, both online and physical, and depending upon the profile of our client-partner and our business relationship we can open the business account in a physical bank.

If you have the money in your company's bank account in the US then you have the right to invest the money however you choose to as long as there is a direct benefit to your business.
Now, if you have the money in your personal bank account and not the business bank account then you can invest the money in real estate, stocks, etc.