In general, is there a relationship between forming an LLC and the need to obtain a business license? Does one get a business license in the state where the LLC will live?
This is a difficult question and the best answer is that there is no static answer. Whether or not a business is required to obtain a business license can depend on the specific jurisdiction in which the entity is filed or does business in as well as the type of business being conducted as well as other factors (number of employees, gross income, etc.).
In some states, counties, or cities, you may be required to obtain a business license even if you are only a sole-proprietor without a corporation or LLC. Sometimes one county in the a state will require a license and another one will not.
I wish i could provide a more specific answer for you, but there are literally thousands of possible licenses in different jurisdictions. Additionally, often there are a myriad of different exemptions for certain businesses or industries, for example some jurisdictions do not require film production companies to obtain business licenses as an incentive to promote filming in their jurisdictions.
I'm sorry I can't provide a more specific answer that would be under 10,000 pages. The best way to get the answer to your question is to contact the local tax authority in the jurisdiction where you conduct business and ask them what the licensing requirements are for your type of business!