Taking your last question first, if your grandmother doesn't want you to do business out of her home, you must find another location.
If your grandmother is willing to allow your corporation to conduct business from her basement, it can do so, subject to limits imposed by the applicable zoning ordinance. Assuming that your grandmother's house is in a residential neighborhood, the zoning ordinance will impose limits on the types of businesses that can be conducted. If your corporation is being noisy, attracting a large number of customers or doing other things that are objectionable to the neighbors, the local zoning administrator may get involved. As to the tax implications, you an pay rent, your grandmother can report it as income and then structure her income tax filing accordingly.
You haven't identified the other parties involved in this question so I cannot determine whether I may have a conflict in this matter. Should it turn out that I have an attorney-client relationship with any of the other parties, my response to this question will not prevent me from continuing to represent an existing client.
Of course your grandmother will have to agree to any use of her property and any rent would be considered income. Your main problem may, or may not, be the applicable zoning rules and regulations. Many cities impose restrictions on at-home businesses.
DISCLAIMER: This answer is provided for general informational purposes only, is not intended as legal advice, and does not create an attorney-client relationship. Undisclosed information will alter any answer provided. Do not rely on this information – it is a starting point for your issue and you should contact a licensed attorney in your jurisdiction.