Cardiac electrophysiology simulations at the cellular level (EMI) are crucial for understanding arrhythmias and developing effective treatments. These large-scale problems, governed by elliptic and parabolic equations, require efficient parallel preconditioners, particularly within domain decomposition frameworks. Moreover, the localized nature of solution features in cardiac electrophysiology simulations necessitates adaptive methods. However, traditional mesh refinement and coarsening techniques often incur significant computational overhead. In this study, we investigate the Balancing Domain Decomposition by Constraints (BDDC) preconditioner combined with algebraic adaptivity and data compression techniques, focusing on their impact on efficiency, convergence, and accuracy in massively parallel computations. Algebraic adaptivity is employed to refine the selection of degrees of freedom, enabling localized resolution of solution features while reducing the computational burden of global solves. By integrating this adaptivity with BDDC, we develop a subdomain-wise adaptive strategy that minimizes overhead while maintaining preconditioner effectiveness. However, the BDDC preconditioner requires expensive reconstruction when the basic time step or degree-of-freedom set changes. We propose a novel combination of BDDC and algebraic adaptivity, demonstrating computational efficiency through numerical examples. This approach employs subdomain-wise subset selection, enabling efficient integration with BDDC while incurring negligible overhead. To address communication bottlenecks inherent in distributed solvers such as BDDC, we incorporate lossy compression, including transform coding and entropy coding, to reduce the volume of data exchanged between subdomains. Our study evaluates the impact of these methods on convergence rates, computational efficiency, and solution accuracy, particularly as the number of subdomains increases. The interplay between algebraic adaptivity and compressed communication is analyzed through numerical experiments, highlighting scenarios where the combined approach achieves significant performance improvements without compromising accuracy. While this approach introduces computational overhead, latency, and potential quantization errors, we present accuracy models and identify cases where lossy compression proves advantageous, supported by practical numerical experiments.