In this paper, the performance of a power domain downlink multiple-input multiple-output non-orthogonal multiple access system in dual-hop full-duplex (FD) relaying networks is investigated over Nakagami-$m$ fading channels by considering the channel estimation error and feedback delay. Particularly, in the investigated system, the base station equipped with multiple antennas transmits information to all mobile users by applying conventional transmit antenna selection/Alamouti-space-time block coding scheme with the help of a dedicated FD amplify-and-forward relay. The received signals at mobile users are combined according to maximal-ratio combining technique to exploit benefits of receive diversity. In order to demonstrate the superiority of the proposed system, outage probability (OP) is investigated and tight lower bound expressions are derived for the obtained OP. Moreover, asymptotic analyses are also conducted for ideal and practical conditions to provide further insights about the outage behavior in the high signal-to-noise ratio region. Finally, theoretical analyses are validated via Monte Carlo simulations and software defined radio based test-bed implementation.