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What kind of business do you plan on opening? How much will you invest? How many people will you employ? Do you have a business in your home country?
So long as you are not working for the entity, you can open the business. The form can be a corporation or a limited liability company. You may want to use a corporate form and have the shares owned by your overseas company. In the future you may be able to transfer employees between the two companies. If someone works for the overseas company for at least a year, they can transfer on a L-1 visa.
If you just want to be able to obtain profits after the employees are paid, you can own the shares directly as a sub chapter S corporation or a limited liability company.
This communication is not intended as legal advice. A local attorney should always be consulted for legal advice. No client/attorney relationship is intended or created by this communication.
To open the business in the US you will need someone who is a US permanent resident or citizen otherwise they will not be able to work in the business. Without the overseas company you will not be able to get a L-1 visa. If you have a subchapter S corporation or an LLC you can receive the profits of the business without working in the business.
Once you get to $100,000 invested you can consider a E-2 treaty investor visa since Turkey is an E-2 visa country. This may be the easiest solution. Sometimes a smaller investment will be sufficient but you can try to generate this much in reserve capital while you are in the US on your masters.
You will not get an H-2 visa because there are plenty of Americans to fill the position. A J-1 visa is for an approved exchange program so you will not qualify for that. Since you can't work how will the business operate. You have to hire someone to work. That person will need to be local. Your overseas company is of no use since it does not own the company you are about to open. Also, you will not get approval for what is essentially a mom and pop size operation.
Yes you share the profits with your overseas company. The simplest way is to have them be shareholders in a LLC, that avoids paying taxes on dividends. To try to bring in a foreign national you will need an immigration lawyer. Some immigration lawyers may also be able to advise on business issues. We cannot make specific recommendations of attorneys but you can locate attorneys in your area on lawyers.com or findlaw.com where they are listed by specialty and geographic area.