Coordinated multi-satellite (CoMS) transmission and non-orthogonal multiple access (NOMA) are envisioned to jointly enhance coverage, capacity, and spectrum efficiency for satellite networks. Their integration into a unified CoMS-NOMA framework will allow more efficient, reliable, and energy-efficient multi-user access. This paper investigates the downlink performance of CoMS-NOMA networks from a system-level perspective, in which multiple satellites cooperatively serve multiple users via NOMA. Leveraging tools from stochastic geometry, related angles and distances in CoMS-NOMA are first derived as intermediate results. Then, we obtain the combined signal power distributions and analyze coverage and spectrum performance under both inter- and intra-satellite interference, accounting for potential imperfect successive interference cancellation (SIC). The analytical model is validated across a range of system parameters, including the number of satellites, service region angle, error-propagation factor, and power allocation coefficients. Numerical results indicate that increasing the number of cooperative satellites does not always improve coverage and spectrum efficiency. Additionally, while a higher main-lobe gain improves coverage, a near-perfect SIC provides only slightly greater benefits than a reasonably good SIC. With properly selected power allocation coefficients, CoMS-NOMA achieves up to a 270% improvement in coverage and a 56% gain in sum spectral efficiency, compared with conventional orthogonal and single-satellite schemes, indicating potential for green, energy-efficient satellite networking.