In this paper, we investigate the diversity, capacity and complexity issues of cooperative Zero-Padding (ZP)-Orthogonal FrequencyDivisionMultiplexing (OFDM) communication. We consider cooperative ZP-OFDM communication over a multipath Rayleigh channel and with multiple Carrier Frequency Offsets (CFOs) existing at different relays. We use a cooperative tall Toeplitz scheme to achieve full cooperative and multipath diversity, while simultaneously combat the CFOs. Importantly, this full diversity scheme only requires Linear Equalizers (LEs), such as Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) equalizers, an issue which reduces the system complexity when compared to a Maximum-Likelihood Equalizer (MLE) or other near-MLEs. Theoretical analysis of the proposed cooperative tall Toeplitz scheme is provided on the basis of the analytical upper bound of the channel orthogonality deficiency derived in this paper. Utilizing only low-complexity linear equalizers, theoretical analysis and simulation results show that the proposed Toeplitz scheme achieves the full cooperative, multipath and outage diversity.