This paper investigates an analytical model for low-earth orbit (LEO) multi-satellite downlink non-orthogonal multiple access (NOMA) networks. The satellites transmit data to multiple NOMA user terminals (UTs), each employing successive interference cancellation (SIC) for decoding. Two ordering schemes are adopted for NOMA-enabled LEO satellite networks, i.e., mean signal power (MSP)-based ordering and instantaneous signal-to-inter-satellite-interference-plus-noise ratio (ISINR)-based ordering. For each ordering scheme, we derive the analytical expression for the coverage probability of each typical UT. Moreover, we discuss how coverage is influenced by SIC, main-lobe gain, and tradeoffs between the number of satellites and their altitudes. Additionally, two user fairness-based power allocation (PA) schemes are considered, and PA coefficients with the optimal number of UTs that maximize their sum spectral efficiency (SE) are studied. Simulation results show that there exists a maximum effective signal-to-inter-satellite-interference-plus-noise ratio (SINR) threshold for each PA scheme that ensures the operation of NOMA in LEO satellite networks, and NOMA provides performance gains only when the target SINR is below a certain threshold. Compared with orthogonal multiple access (OMA), NOMA increases UTs' sum SE by as much as 35%. Furthermore, for most SINR thresholds, the sum SE increases with the number of UTs to the highest value, whilst the maximum sum SE is obtained when there are two UTs.