This study introduces a multiscale simulation framework, termed Projection-based Embedded Discrete Fracture Modeling with Algebraic Dynamic Multilevel method (pEDFM-ADM), which integrates an embedded discrete fracture network representation with a fully algebraic, front-tracking-based mesh adaptation strategy. Incorporating a fully implicit scheme, compositional thermodynamics, and algebraic multilevel operators, the framework captures essential subsurface processes such as buoyancy-driven migration, convective dissolution, phase partitioning, and fracture-matrix interactions under geologically realistic conditions. The method constructs a hierarchy of multilevel grids and localized multiscale basis functions that introduce fine-scale heterogeneities at each coarse level. Adaptive mesh refinement and coarsening are driven by local variations in CO₂ mass fraction and executed through algebraic prolongation and restriction operators, enabling efficient projection between grid levels. The framework is systematically evaluated across a sequence of test cases with increasing complexity, including systems with low-permeability flow barriers, highly conductive fractures, striking a trade-off between computational resource and detailed simulation accuracy. Overall, the pEDFM-ADM framework provides a scalable, fully algebraic, and physically adaptive modeling tool for large-scale CO₂ storage simulations in fractured porous media, supporting predictive simulation and risk assessment for long-term carbon sequestration.