The growing demands of ubiquitous and resilient global coverage have pushed existing networks to their operational limits, making it increasingly difficult to meet all requirements on their own. Integrating \emph{Terrestrial Base Stations (TBS), High Altitude Platform Stations (HAPS)} and \emph{Low-Earth-Orbit (LEO)} satellites is envisioned as a promising solution, yet the coordination across these heterogeneous platforms remains an open challenge. This paper proposes a novel unifying \emph{Triple-C framework: Cooperation, Complementarity, and Competition}, that systematically defines the TBS-HAPS-LEO interaction to deliver seamless resilient and scalable connectivity. For each C, we detail the architectural methodology, required pre-requisites, and measurable deliverables that govern when and how the three layers should collaborate, complement each other, or contend. We further identify the enabling technologies across physical, logical, and cognitive layers to operationalize the proposed 3C paradigm. A rich portfolio of use cases and targeted applications demonstrates how this technological leap will make such integration both feasible and impactful. Comprehensive performance analysis and emulation results quantify the trade-offs of such integrated networks. In addition, we examine the economical, environmental, safety, privacy, standardization, and regulatory implications that shape the real-world implementation of the proposed framework. eventually, we provide the gap analysis, outline key technical/non-technical challenges, and a road-map of future research directions needed to unlock the full potential of Cooperation, Complementarity, and Competition operations in TBS-HAPS-LEO integrated networks.