This paper investigates the performance of an intelligent reflecting surface (IRS)-assisted terahertz (THz) communication system, where the IRS facilitates connectivity between the source and destination nodes in the absence of a direct transmission path. The source-IRS and IRS-destination links are subject to various challenges, including atmospheric attenuation, asymmetric $α$-$μ$ distributed small-scale fading, and beam misalignment-induced pointing errors. The IRS link is characterized using the Laguerre series expansion (LSE) approximation, while both the source-IRS and IRS-destination channels are modeled as independent and identically distributed (i.i.d.) $α$-$μ$ fading channels. Furthermore, closed-form analytical expressions are derived for the outage probability (OP), average channel capacity (ACC), and average symbol error rate (ASER) for rectangular QAM (RQAM) and hexagonal QAM (HQAM) schemes over the end-to-end (e2e) link. The impact of random co-phasing and phase quantization errors are also examined. In addition to the theoretical analysis, deep neural network-based frameworks are developed to predict key performance metrics, facilitating fast and accurate system evaluation without computationally intensive analytical computations. Moreover, the asymptotic analysis in the high-signal-to-noise ratio (SNR) regime yields closed-form expressions for coding gain and diversity order, providing further insights into performance trends. Finally, Monte Carlo simulations validate the theoretical formulations and present a comprehensive assessment of system behavior under practical conditions.