In decode-and-forward (DF) relaying networks, the received signal at the destination may be affected by multiple impairments such as multiple channel gains, multiple timing offsets (MTOs), and multiple carrier frequency offsets (MCFOs). This paper proposes novel optimal and sub-optimal minimum mean-square error (MMSE) receiver designs at the destination node to detect the signal in the presence of these impairments. Distributed space-time block codes (DSTBCs) are used at the relays to achieve spatial diversity. The proposed sub-optimal receiver uses the estimated values of multiple channel gains, MTOs, and MCFOs, while the optimal receiver assumes perfect knowledge of these impairments at the destination and serves as a benchmark performance measure. To achieve robustness to estimation errors, the estimates statistical properties are exploited at the destination. Simulation results show that the proposed optimal and sub-optimal MMSE compensation receivers achieve full diversity gain in the presence of channel and synchronization impairments in DSTBC based DF cooperative networks.