We present a general theory for semiconductor polarons in the framework of the Frohlich interaction between electrons and phonons. The latter is investigated using noncommuting phonon creation/annihilation operators associated with a natural set of nonorthogonal modes. This setting proves effective for mathematical simplification and physical interpretation and reveals a nested coupling structure of the Frohlich interaction. The theory is nonperturbative and well adapted for strong electron-phonon coupling, such as found in quantum dot (QD) structures. For those particular structures we introduce a minimal model that allows the computation and qualitative prediction of the spectrum and geometry of polarons. The model uses a generic nonorthogonal polaron basis called "the natural basis." Accidental and symmetry-related electronic degeneracies are studied in detail and are shown to generate unentangled zero-shift polarons, which we consistently eliminate. As a practical example, these developments are applied to realistic pyramidal GaAs QDs. The energy spectrum and the three-dimensional geometry of polarons are computed and analyzed, and prove that realistic pyramidal QDs clearly fall in the regime of strong coupling. Further investigation reveals an unexpected substructure of "weakly coupled strong coupling regimes," a concept originating from overlap considerations. Using Bennett's entanglement measure, we finally propose a heuristic quantification of the coupling strength in QDs.