In this paper, we study a distributed time-reversal space-time block coded single-carrier (D-TR-STBC-SC) system for amplify-and-forward (AF) half-duplex relaying in frequency-selective Rayleigh fading channels. Under the imperfect channel estimation condition, we analyze the mean-square-error (MSE) performance of the optimal and channel-mismatched frequency domain minimum MSE (FD-MMSE) and least square (LS) equalization. Our analysis results show that, unlike the point-to-point communications, the channel-mismatched FD-MMSE equalization of D-TR-STBC-SC relaying network leads to the ceiling effect that the MSE increases as the signal-to-noise ratio (SNR) of relay-to-destination link increases. Decomposing the MSE, it is found that the primary cause of the ceiling effect is the source-to-destination link in the first time-slot, which makes the covariance matrix of noise vector ill-conditioned. In order to resolve the channel-mismatching problems in the equalization process, we develop optimum relay power control strategies by considering practical channel estimations, i.e., training-based LS and linear minimum MSE (LMMSE) channel estimations. It is shown that the optimum power control resolves the trade-off between MSE performance and relay power consumption, and improves the robustness against the channel-mismatching. Finally, we introduce a performance evaluation to demonstrate the performance of channel equalization combined with the proposed power controls in D-TR-STBC-SC relaying network.