This paper analyzes the performance of Alamouti coded short-packet non-orthogonal multiple access (NOMA) systems with hybrid multicast and unicast transmission over Nakagami-m fading, where only the statistical channel state information is available at the transmitter, and the multicast and unicast signals are intended for all users and a particular user, respectively. Due to some practical limitations, channel estimation errors (CEE) caused by a linear minimum mean-square error estimator and residual hardware impairments (RHI) caused by imperfect hardware implementation are taken into account. We first derive the approximate closed-form expressions for the average block error rate (BLER) and the corresponding asymptotic expressions at all users. Using such expressions, we analyze the diversity performance including conventional diversity order and finite-SNR diversity order. After this, we quantify the relationship among the common blocklenth, power allocation, and pilot sequence length under users' reliability constraints. Finally, numerical and simulation results verify our theoretical analysis and show that i) CEE and RHI greatly affects the transmission blocklength, ii) RHI leads to the error floor phenomenon at high SNRs and the finite-SNR diversity order is the effective performance metric in the presence of RHI, iii) there exist the optimal power allocation coefficients and pilot sequence length to minimize the transmission blocklength in the considered hybrid multicast-unicast system, and iv) the NOMA scheme is superior to the OMA counterpart by achieving low-latency transmission.