Idaho "corporation", "incorporated", "company", "limited", or the abbreviation "corp.", "inc.", "co.", or "ltd.", or words or abbreviations of like import in another language; provided however, that if the word "company" or its abbreviation is used it shall not be immediately preceded by the word "and" or by an abbreviation of or symbol representing the word "and" § 30-1-401 Idaho Statutes
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.
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.
Where the equipment you use in your business was acquired as a gift, you may report your estimate of its current market value on the Business Property Statement (that is, what you think it would sell for in the open market place). Enter that estimated value in the most current year's cost line and add a note indicating that the entry is an estimate.