The estimation of the performance of terahertz wireless communication under the effect of various weather conditions is vital. In this work, the combined effects of cold weather conditions such as snow, as well as random effects of turbulence and pointing errors (PEs) between the transmitter and receiver, on the performance of terahertz wireless line-of-sight links have been evaluated. The exponentiated Weibull distribution has been employed to derive exact analytical closed-form expressions in the presence of combined channels. Our predictions indicate that snow can have various influences on the average bit error rate (BER) and the average channel capacity of terahertz wireless links. Dry and wet snow, based on its liquid-water content, shows different effects in terms of link parameters such as distance and frequency compared to that of rain. Random turbulence and PEs further deteriorate the link performance. It is concluded that the channel capacity is less affected than the BER under the effects of turbulence and PEs, while snow can have a remarkable effect. All of the predictions through the derived expressions are validated using Monte Carlo simulations.