In this paper, we analyze the performance degradation of a multi-hop decode-and-forward full-duplex relaying (MH-DF-FDR) system caused by the residual self-interference (RSI) and hardware distortions (HWD) imposed by the FDR operation and imperfect hardware, respectively. In addition, we study the benefits of employing improper Gaussian signaling (IGS) in the MH-FDR system. Different from the traditional symmetric signaling scheme, i.e., proper Gaussian signaling (PGS), IGS has non-zero pseudo-variance that can limit the impact of RSI and HWD in the MH-FDR system. To evaluate the system performance gain using IGS, first we express the end-to-end achievable rate of the MH system as the minimum rate supported by all participating links. Then, we optimize the pseudo-variance of all participating transmitters including source and relays to compensate the interference impact and improve the end-to-end achievable rate. We propose two network optimization schemes based on the system characteristics i.e. joint optimization framework and distributed optimization scenario. Interestingly, IGS-based scheme outperforms its counterpart PGS-based scheme, especially at higher interference-to-noise ratio. Our findings reveal that using IGS in single-user detection systems that suffer from both RSI and HWD can effectively mitigate the degradation in the achievable rate performance.