A company that I'm working for is registered as a UK limited company.
Anyone can open an LLC in any state, including non-resident aliens. You can go online to file the paperwork for a fee. Each state has its own fee structure and reporting requirements. For example, you will need to pay a minimum of $800 in franchise taxes annually in California.
After you set up the company, you will need to apply for a Federal Taxpayer Identification Number (Form SS-4). You can open a bank account after you have an tax number. The LLC is a legal entity and so you, the non-resident alien, should also be exempt from the non-resident income tax withholding on any interest earned on that account.
If you choose a single-member LLC, the parent would have to file in the US. If the parent is a foreign corporation, the parent has to file not the branch. If the parent is an individual, they would have to file Form 1040NR and Schedule C.
There are also state tax issues you need to consider. Some states, such as NY requires the foreign company to report worldwide income and then apportions that income to the US entity based on the appropriate percentage.
You need a US tax professional to advise you on the pitfalls of this strategy.
Having helped close to 20 European companies set up operations in the U.S., and having answered similar questions many times, I offer the following:
You are right to want to set up a business entity in the U.S. to act as a legal firewall between the subsidiary's and the parent's respective operations. Please see Should I form an LLC or a corporation? for some thoughts about the type of entity to form.
Obtaining an federal Employer Identification Number can be rather tricky if the person running the business does not already have a social security number. Please see Foreign Company Alert: Obtaining an EIN.
It might take several months to obtain the desired work visa. Please see Visa Basics for Foreign Entrepreneurs Coming to the U.S.
You should look for a U.S.-based accountant who is familiar with international tax issues.
Disclaimer: This post does not constitute legal advice and does not establish an attorney-client relationship.
Responding to the issues raised by waxingibbous:
LLC vs. S corporation - Please see Should I form an LLC or a corporation? Which state - Please see Why (not) Incorporate in Delaware (discusses California because that is where I am, but can apply equally well to other states).
No, you don't "just write a check". Proper internal documentation and, possibly, state filing must take place.
You would have a branch if you did not create a separate U.S. entity (corporation or LLC). If the parent creates, and buys the equity interest in, a separate U.S. entity, that entity is a subsidiary. Principal (highly-motivating) reason for creating a subsidiary: If properly formed and maintained, creates a legal firewall - U.S. debts and obligations stay in the U.S. and are not the responsibility of the overseas parent. (Given the litigious nature of the U.S., it probably would be foolish not to create a separate U.S. entity.)
Disclaimer: This post does not constitute legal advice and does not establish an attorney-client relationship.