Generally, once you form a company and file your company name with your state's Secretary of State, you can use that company name where ever you like across the globe.
But that's different from filing a trademark. A trademark typically protects brand names and logos used on goods and services. Your company name is not necessarily a brand name -- and not all company names can get trademark protection.
For example, if your company name is "Online Film Productions," the Trademark Office may well consider that far too basic to qualify for protection, as it merely describes what you do.
In addition, receiving a U.S. trademark does not by itself qualify you for protection around the world. There are additional steps you need to take to register your trademark in accordance with various international treaties.
As an Internet-based film production company, you'll be facing a number of legal issues: 1. As you pointed out, national and possibly international trademark issues; 2. International copyright issues -- and how to prevent others from pirating your work; and 3. Various contract issues to cover the freelancers or other companies to whom you outsource aspects of film creation and production.
You should discuss these issues with an attorney or firm that deals with film production and international trademark issues. They should help give you clear guidance on the best next steps for your business.
Related: Protect Yourself
Related: Don't File for That Patent Yet
Nina L. Kaufman, Esq. is an award-winning New York City attorney, edutainer and author. Under her Ask The Business Lawyer brand, she reaches thousands of entrepreneurs and small business owners with her legal services, professional speaking, information products, and LexAppeal weekly ezine. She also writes the Making It Legal blog.