While device-to-device (D2D) communication between two proximate cellular terminals improves spectral efficiency, cooperative spectrum sharing (CSS) among D2D users (DUs) and cellular users (CUs) provides more gains. In this paper, we thus propose a new CSS scheme, where a pair of DUs (D1 and D2) acts as half-duplex two-path successive relays (TPSRs) between two CUs (C1 and C2) while maintaining cellular full-duplex gains. The channel between D1 and D2, referred to as TPSR interrelay interference channel, is exploited to provide D2D access with an ideal throughput of 1 symbol per time slot (sym/TS). To eliminate the interference between C1-C2 and D1-D2 links, we propose a hybrid complex field network coding (HCFNC) scheme for two cases: First, both of D1 and D2 correctly decode cellular data; and second, either D1 or D2 correctly decodes cellular data. For each case, we derive the closed-form CU diversity-multiplexing tradeoff (DMT) and data rate, symbol error probability for the HCFNC strategy, and the D2D throughput. Our performance analysis and simulation results show that: first, the proposed scheme achieves the 3 x 1 multiple-input single-output DMT; second, the CU achievable rate approaches the full-duplex upper bound; and third, the DU average throughput approaches 1 sym/TS at high signal-to-interference-plus-noise ratio, demonstrating that DUs gain full access opportunity to the cellular spectrum.