We provide in this paper a comprehensive solution to the design, performance analysis, and optimization of a multi-antenna non-orthogonal multiple access (NOMA) system for multiuser downlink communications under a general limited channel state information (CSI) feedback framework for frequency division duplex mode. We design a general framework including user clustering, joint power and bits allocation, CSI quantization and feedback, signal superposition coding, transmit beamforming, and successive interference cancellation at receivers. Then, we conduct a mathematically strict performance analysis of the considered system, and obtain a closed-form lower bound on the ergodic rate of each user in terms of transmit power, CSI quantization accuracy and channel conditions. For exploiting the potentials of multiple-antenna techniques in NOMA systems, we jointly optimize two key parameters, i.e., transmit power and the number of feedback bits allocated to each user, and propose low-complexity closed-form solutions. Moreover, through asymptotic analysis, we reveal the interactions between the main system parameters and their impacts on the joint power and feedback bits allocation result, and hence show some guidelines on the system design. Finally, numerical results validate the correctness of our theoretical analysis and demonstrate the advantages of the proposed algorithms over the most related state of the art.