I''m not a US citizen. Is it legal to run E-Commerce business across the states? Do I need to register for business license and pay tax?

As a general matter, a non-citizen of the United States has all of the right to operate an e-commence business selling to residents of the United States as a citizen. Whether you need to be licensed and pay U.S. taxes depends on where you are located. Since your inquiry suggests you are located in the District of Columbia, then you would be required to pay U.S. taxes on your profits. As a general matter, an unincorporated business does not need a license, but the answer to that question may depend on the precise nature of your business. See an attorney.