From 5G onwards, Non-Terrestrial Networks (NTNs) have emerged as a key component of future network architectures. Leveraging Low Earth Orbit (LEO) satellite constellations, NTNs are capable of building a space Internet and present a paradigm shift in delivering mobile services to even the most remote regions on Earth. However, the extensive coverage and rapid movement of LEO satellites pose unique challenges for NTN networking, including user equipment (UE) access and inter-satellite delivery, which directly impact the quality of service (QoS) and data transmission continuity. This paper offers an in-depth review of advanced NTN management technologies in the context of 6G evolution, focusing on radio resource management, mobility management, and dynamic network slicing. Building on this foundation and considering the latest trends in NTN development, we then present some innovative perspectives to emerging challenges in satellite beamforming, handover mechanisms, and inter-satellite transmissions. Lastly, we identify open research issues and propose future directions aimed at advancing satellite Internet deployment and enhancing NTN performance.