Interfacial transport processes in two-phase flows, e.g., thermocapillarity, boiling, condensation and mass transfer, are frequent in nature and industry. Multiple examples can be found in thermal energy technology, chemical reactors, and the so-called unit operations of chemical engineering. This work presents further advances on the Unstructured Conservative Level-Set (UCLS) method for two-phase flows with complex interfacial physics, i.e, Marangoni effects, liquid vapor phase change and interfacial mass transfer. This method employs a multi-marker UCLS approach to circumvent the numerical coalescence of bubbles and droplets. The finite-volume method discretizes transport equations on 3D collocated unstructured meshes. The fractional-step projection method solves the pressure-velocity coupling in momentum transport equation. The convective term of the momentum transport equation, level-set advection equations, energy equation and mass transfer equation is discretized by unstructured flux-limiters schemes to minimize the so-called numerical diffusion, and to avoid numerical oscillations around the interface. Adaptive mesh refinement (AMR) on collocated unstructured meshes is coupled to the multiphase unstructured solver to optimize computational resources. Such a combination of numerical techniques preserves the numerical stability and accuracy in two-phase flows with complex interfacial physics. Verifications and validations are presented for thermocapillarity migration of droplets, mass transfer, and boiling heat transfer on fixed meshes and adaptive meshes.