This study introduces and investigates the integration of a cell-free architecture with bistatic backscatter communication (BiBC), referred to as cell-free BiBC or distributed access point (AP)-assisted BiBC, which can enable potential applications in future (EH)-based Internet-of-Things (IoT) networks. To that purpose, we first present a pilot-based channel estimation scheme for estimating the direct, cascaded, forward channels of the proposed system setup. We next utilize the channel estimates for designing the optimal beamforming weights at the APs, reflection coefficients at the tags, and reception filters at the reader to maximize the tag sum rate while meeting the tags' minimum energy requirements. Because the proposed maximization problem is non-convex, we propose a solution based on alternative optimization, fractional programming, and Rayleigh quotient techniques. We also quantify the computational complexity of the developed algorithms. Finally, we present extensive numerical results to validate the proposed channel estimation scheme and optimization framework, as well as the performance of the integration of these two technologies. Compared to the random beamforming/combining benchmark, our algorithm yields impressive gains. For example, it achieves $\sim$ 64.8\% and $\sim$ 253.5\% gains in harvested power and tag sum rate, respectively, for 10 dBm with 36 APs and 3 tags.