Is the scenario of "an employer who owns a factory but not its capital, and has two individuals who deposit their money with him for investment, and they receive their share of the profit based on the capital, while the employer only receives a monthly salary" considered a Mudarabah (profit-sharing partnership) or an Ijarah (lease/hire)? If it is valid as an Ijarah, is it permissible for the employee to invest his own money as part of the capital and receive profit from it? And is it permissible for a third party to join this arrangement?

There is no impediment for the factory owner to rent out himself and his factory to capital owners for a known monthly wage, provided that the rulings of ijarah (definition) (leasing/hiring) are applied, which include knowledge of the work and the hirer not being liable except in cases of transgression or negligence. It is also permissible for the factory owner (the ajir or hired party) to agree with the capital owners to participate with them in the capital, whereby he receives a share of the profit according to his stake, in addition to his wage for his work and his factory.