The interplay between periodic potentials and artificial spin-orbit coupling (SOC) in ultracold atomic gases provides a versatile platform for exploring exotic quantum phases. In contrast to conventional optical lattices based on Raman coupling modulation, a Zeeman lattice characterized by a spatially varying longitudinal field (σ_z modulation) exhibits distinct symmetry properties and competing mechanisms. Here, we systematically investigate the ground-state phase diagram and elementary excitations of a Ramaninduced SOC Bose-Einstein condensate in a one-dimensional Zeeman lattice. At the single-particle level, we show that the Hamiltonian possesses a composite symmetry—combining PT and extended reflection symmetries—which protects a robust band degeneracy at the Brillouin zone boundary. Driven by the competition between the periodic Zeeman splitting and Raman-induced spin mixing, the quasi-momentum minimum displays a non-monotonic re-entrant evolution across different momentum regimes as the Raman coupling strength varies. Incorporating many-body interactions, we construct the ground-state phase diagram using a variational ansatz and the Gross-Pitaevskii equation, identifying four distinct phases: the band-edge, single-momentum Bloch, stripe, and band-center states. The phase transitions are found to be predominantly second-order and strongly governed by the underlying band structure and symmetry constraints. We further analyze the elementary excitation spectra within the Bogoliubov-de Gennes framework and compute the static density and spin structure factors. The results reveal characteristic dynamical signatures, including an inversion of the density and spin excitation branches in the band-edge state, as well as distinct crossing features in the Bloch state linked to the single-particle dispersion minima. Our findings elucidate the critical role of symmetry and band structure in shaping the quantum phases and collective excitations of SOC BECs in Zeeman lattices, providing clear experimental signatures accessible via Bragg spectroscopy and spin-resolved measurements.