In this paper, we propose a distributed flexible coupler antenna (FCA) array to enhance communication performance with low hardware cost. At each FCA, there is one fixed-position active antenna and multiple passive couplers that can move within a designated region around the active antenna. Moreover, each FCA is equipped with a local processing unit (LPU). All LPUs exchange signals with a central processing unit (CPU) for joint signal processing. We study an FCA-aided multiuser multiple-input multiple-output (MIMO) system, where an FCA array base station (BS) is deployed to enhance the downlink communication between the BS and multiple single-antenna users. We formulate optimization problems to maximize the achievable sum rate of users by jointly optimizing the coupler positions and digital beamforming, subject to movement constraints on the coupler positions and the transmit power constraint. To address the resulting nonconvex optimization problem, the digital beamforming is expressed as a function of the FCA position vectors, which are then optimized using the proposed distributed coupler position optimization algorithm. Considering a structured time domain pattern of pilots and coupler positions, pilot-assisted centralized and distributed channel estimation algorithms are designed under the FCA array architecture. Simulation results demonstrate that the distributed FCA array achieves substantial rate gains over conventional benchmarks in multiuser systems without moving active antennas, and approaches the performance of fully active arrays while significantly reducing hardware cost and power consumption. Moreover, the proposed channel estimation algorithms outperform the benchmark schemes in terms of both pilot overhead and channel reconstruction accuracy.