Existing research works on reconfigurable intelligent surfaces (RIS) based terahertz (THz) system ignores the effect of phase noise and employ the zero-boresight pointing errors model of the free-space optics channel in performance analysis. In this paper, we analyze the performance of RIS-THz transmission under the combined effect of channel fading, THz pointing error (TPE), and statistical phase noise due to imperfect phase compensation at each RIS element. First, we derive statistical results of the double $α$-$μ$ fading combined with the TPE and phase noise at individual RIS elements using single-variate Fox's function. Next, we use the multi-variate Fox's H-function representation to develop exact analytical expressions for the density and distribution functions of the resultant signal-to-noise ratio (SNR) of the RIS-THz link considering the accumulating propagation effect from all RIS elements. Using the derived statistical results, we analyze the exact and asymptotic expressions for the considered system's outage probability and average bit-error rate (BER). The analytical results show that the diversity order of the system is independent of phase noise, depends on the channel fading parameters $α$ and $μ$, and depends on the $β$ parameter of the TPE.