GeSe monolayer (ML) has recently attracted much interest due to its unique structure and excellent physical properties that can be effectively tuned through single doping of various elements. However, the co-doping effects on GeSe ML are rarely studied. In this study, the structures and physical properties of Mn-X (X = F, Cl, Br, I) co-doped GeSe MLs are investigated by using first-principle calculations. The formation energy and phonon disspersion analyses reveal the stability of Mn-Cl and Mn-Br co-doped GeSe MLs and instability of Mn-F and Mn-I co-doped GeSe MLs. The stable Mn-X (X = Cl, Br) co-doped GeSe MLs exhibit complex bonding structures with respect to Mn-doped GeSe ML. More importantly, Mn-Cl and Mn-Br co-doping can not only tune magnetic properties, but also change the electronic properties of GeSe MLs, which makes Mn-X co-doped GeSe MLs indirect band semiconductors with anisotropic large carrier mobility and asymmetric spin-dependent band structures. Furthermore, Mn-X (X = Cl, Br) co-doped GeSe MLs show weakened in-plane optical absorption and reflection in the visible band. Our results may be useful for electronic, spintronic and optical applications based on Mn-X co-doped GeSe MLs.