Is there a legal way for a foreigner to start a business in US?

I am on a work visa (H1-B) in the US. From what I understand,
the only way for me to make money is through my employer. I've some
skills that I think I can use to create a product (or website etc).
My question is:

1. Is there any legal way for me to start a company?

2. If a family member of mine has EAD or green card, can he/she
start a company, do all the accounting/legal/taxes etc in their name,
while I take care of the technical stuff. Of course, I remain as
a non-paid part of the company, I don't make a single penny out of
it (which is fine by me. I know it takes a few years to make any serious
amount of revenue anyway).

3. If the above is not possible, can I incorporate a company in my
country, but work from here on my freetime? And pay taxes in my
country.

I know there are other options like the E visa (I think thats the name)
which is given to those individuals who invest $500k or more. I am not
looking into those, as I don't have that much money or expertise or risk
taking ability.

You can easily start a business in the U.S.; a major issue will be that you cannot work for that business here because you will not have the right type of visa (please see Visa Basics for Foreign Entrepreneurs Coming to the U.S. ).

A second challenge will be obtaining an Employer Identification Number (federal taxpayer identification number) - please see Foreign Company Alert: Obtaining an EIN may be your Biggest Challenge in the U.S. Disclaimer: This post does not constitute legal advice and does not establish an attorney-client relationship.

You can be a founder of a company that petitions for your visa.

You'd need to quit the employer that is currently sponsoring your H-1B visa and get a new H-1B sponsored by the company you are founding.

Note that you will probably need to leave the US during this process (I did) and you will need to demonstrate that the new company isn't just a front to keep you in the country (helps if you have other founders, other employees, don't have a controlling interest, etc)