Three-dimensional topological insulators (3D TIs) possess a specific topological order of electronic bands, resulting in gapless surface states via bulk-edge correspondence. Exotic phenomena have been realized in ferromagnetic TIs, such as the quantum anomalous Hall (QAH) effect with a chiral-edge conduction and a quantized value of the Hall resistance Ryx. Here, we report on the emergence of distinct topological phases in paramagnetic Fe-doped (Bi,Sb)2Se3 heterostructures with varying structure architecture, doping, and magnetic and electric fields. Starting from a 3D TI, a two-dimensional insulator appears at layer thicknesses below a critical value, which turns into an Anderson insulator for Fe concentrations sufficiently large to produce localization by magnetic disorder. With applying a magnetic field, a topological transition from the Anderson insulator to the QAH state occurs, which is driven by the formation of an exchange gap owing to a giant Zeeman splitting and reduced magnetic disorder. A topological phase diagram of (Bi,Sb)2Se3 allows exploration of intricate interplay of topological protection, magnetic disorder, and exchange splitting.