The recent advances of high spin-orbit torque efficiency in ferromagnetic/topological insulator (TI) structures hold great promise for the development of high-performance spintronic devices. However, the roles of spin-momentum-locked Dirac surface states (SSs) and the interfacial magnetic proximity effect (MPE) on dynamic magnetic properties are still under debate. Here, we quantitatively distinguish the manipulation effects of SSs and MPE on magnetic damping and surface anisotropy in TI/nonmagnet/CoFe heterostructures. We found that, in addition to the common spin pumping contribution stemming from SSs, damping enhancement also consists of an obvious MPE contribution to the TI/CoFe material system. Moreover, the increased surface magnetic anisotropy for the CoFe films grown on top of a TI layer is believed to arise mainly from the interfacial atomic intermixing. Our paper sheds light on the effects of SSs and MPE on magnetization dynamics, which offer exciting opportunities for developing TI-based spintronics.