Is it permissible for an employee in an institution to participate in a project that he was assigned to supervise in his work, while receiving wages for some of the tasks, and is it obligatory to donate all profits and expenses, or only the profits, in case there is doubt about the occurrence of something prohibited?

The agreement to grant you the project's completion for his institution in exchange for compensation you pay him is forbidden (haram), because the money paid to the employee is a bribe, which is illicit gain (suht) that is prohibited, and it falls under consuming people's wealth unjustly. The Messenger of Allah, peace be upon him, cursed the briber, the bribed, and the facilitator. If you are the most deserving, then the employee should assign the project to you without compensation. If you are not deserving, then this is deception and betrayal. As for the money you earned from working on the project, there is no blame on you for benefiting from it if you completed it as required, but the sin of the bribe is upon you.