This thesis explores the advancement and refinement of numerical models for beams within the finite element method, a critical tool widely adopted across industries for simulations. Specifically, my research has been focused on the dynamics of beams and their conservation properties when considered using geometrically non-linear models. The second research question concerns contacts between beams. Both of these are significant as model reduction techniques, which allow faster and more complex simulations of our physical environment. The core objectives of this work are to investigate the conservation of mechanical properties of beams for long-term stable formulations and to develop a robust method for understanding and simulating beam-to-beam contact mechanics across diverse scenarios. Our methodology blends theoretical analysis with numerical simulations, focusing on geometrically-exact beam theory and beam-to-beam contacts, supported by the development of open-source research code. As a result, I have developed a novel dynamic beam finite element, which is capable of conserving linear and angular momentum and almost conserve energy. This element features an advanced interpolation of position and rotation on SE(3) group, which allows the element to be objective and locking-free. The conservation properties have been achieved using the Lie midpoint rule for time integration and an independent velocity field. Additionally, I have devised several contact formulations for line-to-line contacts, notably applying the existing mortar method to beams, and introducing a novel, unbiased contact formulation. Both of these methods have been combined with the Lagrange-multiplier and the penalty method and compared using benchmark tests. The thesis contributes to the field by offering novel approaches in beam dynamics and contacts, characterised by robustness and stability. I have discussed their advantages and drawbacks in detail and point out potential areas for further investigation.