Your name must be unique, and not deceptively similar, to any other trademarked name or business. It is also required that your name not be used to intentionally misrepresent the products or services you offer. For LLCs, nearly all states will also require you to add a signifier of your limited liability status, such as "LLC" or "L.L.C." to the end of your company's name. You may be able to operate under a name other than your formal LLC name by applying for and using a dba.
If you received a Notice to File a Business Property Statement from the Assessor and your business is no longer in operation, you are still required to file the Business Property Statement. You should also include a note on the Business Property Statement indicating that the business has closed, so the Assessor does not continue to assess the property.
On the other hand, if you are not responsible for its tax, then you should declare the equipment in Part III of the Business Property Statement (Equipment belonging to others). Where equipment is declared in Part III of the Business Property Statement, the Assessor will also send a Notice to File to the person reported as the equipment's actual owner.