We present a generalized theory based on one-dimensional Maxwell-Bloch equations to study the amplification process of an inner-shell photoionization-pumped atomic x-ray laser. Focusing an x-ray free-electron laser beam in an elongated neon-gas target results in a strong exponential amplification of Kα fluorescence, as recently demonstrated [N. Rohringer et al., Nature (London) 481, 488 (2012); C. Weninger et al., Phys. Rev. Lett. 111, 233902 (2013)]. Here, we present an in-depth theoretical study of the amplification process that goes beyond the previous theory based on a rate-equation approach. We study the evolution of the pulse characteristics during the amplification process for transform-limited Gaussian and broadband self-amplified spontaneous-emission pump pulses. We discuss the impact of the gain-dependent group velocity on the emitted x-ray radiation and the resulting gain-guiding effects. A thorough analysis of the spectral and temporal properties of the emitted radiation is presented, including higher-order field-correlation functions, to characterize the ensemble of emitted x-ray pulses.