Simple answer: no
Complicated answer: perhaps ... but, only if an attorney prepares LOTS of papers and the start-up is incredibly well funded.
NOTE: No, you can't work for your own company.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
The boundaries of the program are the same as H1B. You cannot be self employed. The idea was to provide Australian with a way to work here without the need to go thorough the song and dance of the H1B numerical limitations. The treaty did not take into account the nonimmigrant intent issue because they believed most Australians will only want to work here and would be eager to return home after the job opportunity ends. In all other respects, its boundaries are identical with H1B as are those of the TN program by the way.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
In light of the E3 restrictions with self employment, look into the E2. You can establish an umbrella company that may allow you to do all that you are wanting to do. However, you will have to invest in a new entity and you will need the help of an attorney(s) to set it up.
Due to the nature of this forum, Attorney Maria J. Marty does not have all the information required to provide legal advice. Accordingly, her responses on Avvo are intended as general and not legal advice.