In this paper, we propose short-packet communications (SPCs) with non-orthogonal multiple access (NOMA) for coordinated and direct relay transmission (CDRT) Internet-of-Things (IoT) networks to enhance spectrum utilization and system performance. In this system, a source transmits its packets to a near IoT user via direct transmission and a far IoT user via selective relaying transmission under practical conditions of imperfect successive interference cancellation and cochannel interference. We derive the closed-form and asymptotic expressions for the average block-error-rate (BLER), throughput, and the goodput of the considered system over Rayleigh fading channels. To further improve the considered system, we formulate the system throughput maximization problem. Numerical results show that our proposed system outperforms the orthogonal multiple access and conventional NOMA-CDRT ones in terms of both BLER, system throughput, and goodput. In addition, by jointly optimizing power allocation coefficients and channel coding ratios, the system throughput performance is maximized. Furthermore, for a given block-length, the proposed system with short messages achieves superior latency and reliability compared to the long-message one. Monte Carlo simulations are presented to confirm the theoretical analysis.