Can I live in Michigan and legally open a small business(sole
proprietorship) and a business bank account in another state
(Tennessee)?

Hello and thank you for your question.

Yes, you can open and legally operate a sole proprietorship and a
business bank account in another state (Tennessee).

You'll need to obtain a Tennessee business license, in the city and
county where you operate:
For example,
http://www.scoreknox.org/library/businesslicense.htm
http://www.countyclerkanytime.com/forms/BK209--Application%20for%20New%20Business%20Tax%20License.pdf
http://www.bristoltn.org/planning/Web%20Page%20Links/Business%20License%20Appliation%5B1%5D.pdf
http://www.cityofathenstn.com/finance/pdf/BusinessLisenseOnline.pdf

You should phone the town clerk where you'll be operating to get the
necessary form.  Be sure to ask about both a city license and a county
license.

As the instructions to the Michigan income tax return show, you will
have to pay Michigan income tax on what you earn in Tennessee
(Tennessee is not one of the 'reciprocal states.'
http://www.michigan.gov/documents/BOOK_MI-1040_111249_7.pdf

That's it!

Search terms used:
tennessee "business license"
tennessee "application for business license"
michigan "income tax"

Thanks again for letting us help

Google Answers Researcher
Richard-ga