This work conceives dictionary-learning (DL)-based sparse channel estimation schemes for multi-user Terahertz (THz) hybrid MIMO systems incorporating also non-idealities such as hardware impairments and beam-squint effect. Due to the presence of large antenna arrays coupled with frequency selectivity, beam squint effect is significant in THz systems. Moreover, the manufacturing and calibration errors that inevitably arise during the production of antenna arrays result in hardware impairments such as irregular antenna spacing, mutual coupling and antenna gain/phase errors in practical THz systems.To overcome these problems, this work proposes a DL algorithm to determine the best sparsifying dictionary from the acquired observations for a single-carrier frequency domain equalization (SC-FDE)based wideband THz system in the presence of hardware impairments as well as the beam squint effect. The dictionary thus obtained is subsequently employed to exploit the sparsity of the MIMO THz channel toward CSI estimation. Furthermore, the Cramér-Rao lower bound (CRLB) is also derived for the joint DL and CSI estimation algorithm, which acts as a benchmark for the mean-squared error (MSE) performance of the channel estimate obtained. The scheme is also extended to SC-FDE-based wideband THz MIMO systems with multiple antenna users. Simulation results are presented to corroborate our analytical findings and also demonstrate the improved performance with respect to the agnostic scheme that ignores the nonidealities.