We consider network-assisted full-duplex (NAFD) cell-free massive multiple-input multiple-output (CF-mMIMO) systems, where full-duplex (FD) transmission is virtually realized via half-duplex (HD) hardware devices. The HD access points (APs) operating in uplink (UL) mode and those operating in downlink (DL) mode simultaneously serve DL and UL user equipments (UEs) in the same frequency bands. We comprehensively analyze the performance of NAFD CF-mMIMO from both a spectral efficiency (SE) and energy efficiency (EE) perspectives. Specifically, we propose a joint optimization approach that designs the AP mode assignment, power control, and large-scale fading (LSFD) weights to improve the sum SE and EE of NAFD CF-mMIMO systems. We formulate two mixed-integer nonconvex optimization problems of maximizing the sum SE and EE, under realistic power consumption models, and the constraints on minimum individual SE requirements, maximum transmit power at each DL AP and UL UE. The challenging formulated problems are transformed into tractable forms and two novel algorithms are proposed to solve them using successive convex approximation techniques. More importantly, our approach can be applied to jointly optimize power control and LSFD weights for maximizing the sum SE and EE of HD and FD CF-mMIMO systems, which, to date, has not been studied. Numerical results show that: (a) our joint optimization approach significantly outperforms the heuristic approaches in terms of both sum SE and EE; (b) in CF-mMIMO systems, the NAFD scheme can provide approximately 30\% SE gains, while achieving a remarkable EE gain of up to 200\% compared with the HD and FD schemes.