In order for me to use my brandname I invented, in which I'm not going to name because it's not copyright protected yet; but can I just use it on my films without it being a legit business?
Your business name cannot be copyright protected. Legally there can be
several businesses using the same name as long as they are in different
businesses. And yes, you can use the name on your films without it
being a legal business.
A production company is no different than any company. A
manufacturing company buys or rents a building and makes a
product. An auto repair company buys or rents a building and fixes
cars. A "production company" make movies. And they are all started
in the same way.
There is a "small business association" in most cities - there is
one in San Francisco. They can help you with legal matters. The
short version: You will fill out the paperwork, make the required
legal declarations and get to work.
You need to know clearly what your goals are. Do you just want to
put a name on the credits of your movie? Do you need to open a
bank account using the company name?
A Sole Proprietorship:
A sole proprietorship is the simplest form of business where one
individual conducts the business. The business owner is personally
liable for all the obligations of the business.
Check here fo some good info. And
HERE for name availability.
Search
Fictitious Business name California for a TON of information on
registering your name and starting your one business.