Is it permissible to benefit from graduation projects written with pirated software after obtaining the project owner's permission, and is the money earned from them lawful?

The basic principle regarding things is permissibility, and there is no obligation to investigate the legality of the source of funds without a compelling reason; doing so is an blameworthy over-elaboration and delving into matters. Using pirated software to write research papers does not prohibit benefiting from them, nor does it affect the lawfulness of the money earned from them. This might be a form of obsessive doubt (waswasah). Similarly, the use of pirated operating systems by individuals does not necessitate the prohibition of benefiting from their books or curricula obtained through lawful means.