Is it permissible for me to carry out tasks unrelated to the project based on the manager's request and to avoid informing the other partner, and would my salary be considered dubious in this case?

If the manager does not have the right to exploit the worker for his personal work, even during the worker's free time, then it is not permissible for you to engage in the manager's personal work during your official working hours, because it constitutes a waste of the primary work and a dereliction of duty. Your salary would then become unlawful to the extent of the wasted time. However, if the manager is entitled to assign employees to his personal work, then there is no harm in that.