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.
It should also be noted that the state of New York requires limited liability companies to comply with an unusually strict set of publication requirements. In addition to publishing notices in two papers in the county in which your business is forming, you will also be required to provide proof of this to New York’s Department of State within 120 days of becoming officially recognized as a business. Failing to do so can result in suspension of your right to conduct business in the state.
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.