This paper presents an analytical exploration of sixth-generation (6G) satellite–terrestrial integrated networks, focusing specifically on their applications within air mobility operations, such as those involving unmanned aerial vehicles (UAVs). As the integration of satellite and terrestrial networks promises to revolutionize mobile communication by extending coverage and enhancing connectivity, this study delves into two critical aspects: link budget analysis and handover and mobility analysis for UAVs. The link budget analysis assesses the communication requirements necessary to ensure robust and consistent connectivity between satellites and UAVs, accounting for factors such as path loss, antenna gains, and power transmission. Meanwhile, the handover and mobility analysis investigates the challenges and solutions associated with UAVs transitioning between different network nodes and layers in a dynamic aerial environment. This paper utilizes theoretical models and simulations to provide insights into the design and optimization of these networks, aiming to enhance the reliability and efficiency of UAV operations in the context of the emerging 6G landscape. The findings propose not only technological advancements in network architecture but also practical guidelines for the deployment of UAVs in complex environments, marking a significant step toward the realization of a fully integrated, satellite-terrestrial ecosystem.