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.
LP, Limited partnership: a partnership where at least one partner (the general partner, which may itself be an entity or an individual) has unlimited liability for the LP's debts and one or more partners (the limited partners) have limited liability (which means that they are not responsible for the LP's debts beyond the amount they agreed to invest). Limited partners generally do not participate in the management of the entity or its business.
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.