This paper presents a formulation of the Circular Restricted Three-Body Problem (CR3BP) in Birkhoff coordinates, with applications to trajectory design and optimal control near the Earth-Moon system. The dynamical equations are explicitly derived in the Birkhoff framework and validated through numerical comparison with the classical rotating frame formulation. The stability characteristics of periodic orbits around all five Lagrange points are analyzed, highlighting the improved linearization behavior of the Birkhoff formulation near the collinear equilibria. Differential correction techniques are used to compute Lyapunov orbits, which serve as reference trajectories for transfer design. Several optimal control problems are solved, including transfers between Lyapunov orbits near L1 and L2, between distinct periodic orbits near L5, and within the L1 family. Performance comparisons in both coordinate systems reveal that the Birkhoff formulation offers superior numerical stability and supports larger transfer domains near unstable points under equivalent linearizations. These findings demonstrate the formulation's potential for robust cislunar trajectory design and its relevance to future mission planning.