Low-altitude wireless networks (LAWNs) are expected to consist of multi-tier, heterogeneous terrestrial and non-terrestrial devices, where effective coordination is essential to fully unlock the complementary capabilities of diverse systems from different vendors. To address this issue, we propose a novel multi-functional coordination framework that enables seamless cooperation within the LAWN while supporting efficient execution of diverse network functions. In the proposed architecture, each device or infrastructure element is assigned to a specific functional role, namely, edge mobile terminal (E-MT), distributed MT (D-MT), or computing center. E-MTs are equipped with lightweight, independent signal processing and computing capabilities, while D-MTs and the computing center handle regional and global coordination, respectively. To enhance the overall network efficiency, we model the LAWN as a sparse graph, where nodes represent network nodes and edges are defined according to a set of controllable connection rules. This topology-aware (TA) representation allows for efficiently solving various coordination tasks across the network. Numerical results show that the proposed TA coordination framework outperforms baseline approaches that lack topological insights, achieving higher efficiency in multi-task coordination. Finally, we discuss key technical challenges and outline potential solutions for future deployment.