As the core nonlinear element underpinning superconducting electronics, the Josephson junction is characterized by its current-phase relation (CPR), which fundamentally determines the dynamical properties and functional capabilities of superconducting quantum devices. Traditional Josephson junctions typically exhibit a conventional sinusoidal CPR; however, junctions characterized by non-sinusoidal CPR have recently attracted considerable attention due to their distinctive physical properties and promising quantum device applications. In this study, we developed a numerical model tailored specifically for junctions exhibiting non-sinusoidal CPR by integrating experimentally measured current-voltage (I-V) characteristics from Nb/Al-AlO_x/Nb junctions into a resistively and capacitively shunted junction (RCSJ) framework. Leveraging this refined model, we systematically explored the influence of CPR skewness on Josephson junction dynamics. Our results reveal that, in underdamped junctions, the critical current significantly diminishes with increasing CPR skewness, a behavior reminiscent of the tunable critical currents typically observed in DC superconducting quantum interference devices (SQUID). Conversely, in overdamped junctions, the influence of CPR skewness on the I-V characteristics is found to be negligible. However, our numerical simulations under microwave irradiation reveal that nonsinusoidal CPRs readily promote the emergence of half-integer Shapiro steps in overdamped junctions, thereby establishing CPR skewness as a plausible microscopic origin for this phenomenon. In addition, we employed Advanced Design System (ADS) simulations to model nonlinear resonators and DC SQUID circuits, offering a detailed investigation into how nonsinusoidal CPRs modulate the Josephson inductance and magnetic flux response. Our findings reveal that engineering the CPR of Josephson junctions provides substantial flexibility in the design of superconducting qubits, parametric amplifiers, and non-magnetic nonreciprocal devices. This tunability underscores significant opportunities for the development of next-generation superconducting electronic components. Josephson junctions with engineered CPR offer expanded functionality for superconducting quantum technologies. This study shows that tailoring CPR enables enhanced control over the dynamical behavior of junctions, facilitating optimized designs of superconducting qubits, parametric amplifiers, and nonmagnetic nonreciprocal devices.