What are the limits for the Imam and his family to use the mosque's necessities, such as water, cups, pens, and papers, and old, unnecessary items belonging to the mosque that might be discarded, given that the mosque collects donations from brothers in America?

If the Imam of the mosque is responsible for its repair, cleanliness, and expansion, then he is the supervisor (nazir) of the endowment (waqf). In this case, it is permissible for him to benefit from its water and papers in a reasonable manner, based on the Prophet's (peace be upon him) saying regarding the endowment of Umar (may Allah be pleased with him): "There is no sin upon its supervisor to eat from it in a reasonable manner or to feed a friend without accumulating wealth." However, if he is not the supervisor, then he is only entitled to what the endower has permitted for the general worshippers.