From fifth-generation (5G) communication technology onward, non-terrestrial networks (NTNs) have emerged as a key component of future network architectures. Especially through the rise of low-Earth-orbit satellite constellations, NTNs enable a space Internet and present a paradigm shift in delivering reliable services to even the most remote regions on Earth. However, the extensive coverage and rapid movement of satellites pose unique challenges in user equipment access and inter-satellite transmission, impacting the quality of service and service continuity. This paper offers an in-depth review of NTN networking technologies in the context of six-generation (6G) mobile networks evolution, focusing on access management, satellite mobility, and hetero-network slicing. Building on this foundation and considering the latest trends in NTN development, we then present innovative perspectives on emerging challenges, including satellite beamforming, handover mechanisms, and service delivery. Lastly, we identify key open research areas and propose future directions to improve NTN performance and accelerate satellite Internet deployment.