The rapid growth in mobile communication technologies has turned mobile edge computing (MEC) into a paradigm-shifting technology that extends cloud-like capabilities and storage resources to the edge of the network. This allows computation-intensive and latency-sensitive applications to be performed at close proximity to the end-users, thereby overcoming the bottleneck issues of resource-constrained devices. However, ensuring efficient operations in MEC-empowered systems requires intelligent task execution and resource allocation across MEC servers. To this end, MEC-empowered non-terrestrial wireless networks (MeNT-WiN) systems are one of the applications in which deep reinforcement learning (DRL) is seen as a powerful method to enhance the MEC abilities in edge servers and network entities. This paper presents a thorough overview of the applications of DRL in MeNT-WiNs. In particular, it underlines the main contribution of DRL in enhancing the performance of MeNT-WiNs, including unmanned aerial vehicles (UAV) and satellite communications networks. This paper investigates how DRL can meet the unique requirements of MeNT-WiNs by enhancing system efficiency, scalability, and decision-making processes across MEC architectures. First, the article reviews the fundamentals of DRL, it later goes on to discuss its integration with MeNT-WiNs and demonstrates its relevance for the optimization of satellite communications and management of UAV swarms, as well as enhancing connectivity in remote areas. The survey also identifies key challenges for DRL-driven MeNT-WiN systems, such as computational complexity and real-time adaptability, while being scalable. Finally, it discusses future research possibilities, emphasizing the importance of new solutions that integrate DRL with MEC in order to fully exploit the potential of MeNT-WiNs.