Carbon fiber battery electrodes combine mechanical load-bearing capability with electrochemical energy storage in structural batteries, where carbon fibers act as high-performance structural reinforcements and active battery electrodes due to their high specific stiffness, strength, and lithium intercalation capacity, while the surrounding matrix provides mechanical support and serves as the medium for lithium ion transport. The fiber-matrix interface is central to this multifunctionality by providing both adhesion for load transfer and electrochemically active surface area, yet debonding at this interface can compromise structural integrity and alter electrochemical performance. In this study, a three-dimensional (3D) multiphysics framework is developed to investigate fiber-matrix interfacial behavior and damage evolution during galvanostatic cycling and under mechanical loading applied along both the longitudinal and transverse directions of the electrode. The framework captures localized effects near fiber ends that are often neglected in two-dimensional analyses and incorporates lithium concentration-dependent fiber properties. The model predicts that lithiation-induced fiber expansion and transverse tensile loading localize interfacial stresses near fiber ends and promote debonding. In contrast, longitudinal tensile loading has a minimal or even suppressive effect on debonding when it is applied during galvanostatic cycling. Electrochemical impedance spectroscopy (EIS) is simulated to assess the impact of fiber-matrix debonding on the electrode impedance spectra, and the resulting changes in charge transfer behavior due to debonding are analyzed. Consistent with recent experimental observations that an electrolyte-filled interfacial region created by fiber-matrix debonding after cycling provides additional ionic transport pathways in the fully delithiated electrode lamina, effective medium approximations are used to predict the enhancement in ionic conductivity in the electrode lamina as a function of fiber volume fraction, the relative size of the interfacial opening with respect to the fiber radius, and the ionic conductivity ratio between the matrix and the electrolyte-filled interfacial region.