Reza,

You could absolutely open a sister company in the United States, which for example may be owned by you individually, or by your UK company. You may use your home or business address in the UK for the business address with your US company as well.

The costs of setting up a business vary greatly upon state to state, and depending on the type of business you are conducting the company might be required to obtain additional licensing or permits. Feel free to send me an email (my contact information is listed below) with the type of business you are looking to conduct, along with a specific state of interest if you have one, and I would be happy to let you know what the costs, time and procedures involved would be. If you didn’t have a state of registration in mind, our team could definitely help advise which options would fit your business plans the best.

Regarding being able to stay in the United States, that is a question best suited for an attorney who is familiar with immigration laws and requirements.