Space-ground communication systems are important in providing ubiquitous services in a large area. This paper considers the fairness designs under a load-balancing framework with heterogeneous receivers comprising access points (APs) and a satellite. We derive an ergodic throughput of each user in the uplink data transmission for an arbitrary association pattern and imperfect channel state information, followed by a closed-form expression with the maximum-ratio combining and rich scattering environments. We further formulate a generic fairness optimization problem, subject to the optimal association patterns for all the users. Despite the combinatorial structure, the global optimal solution to the association patterns can be obtained by an exhaustive search for small-scale networks with several APs and users. We design a low computational complexity algorithm for large-scale networks based on evolutionary computation that obtains good patterns in polynomial time. Specifically, the genetic algorithm (GA) is adapted to the discrete feasible region and the concrete fairness metrics. We extensively observe the fairness design problem by incorporating transmit power control and propose a hybrid genetic algorithm to address the problem. Numerical results demonstrate that the association pattern to each user has a significant impact on the network throughput. Moreover, the proposed GA-based algorithm offers the same performance as an exhaustive search for small-scale networks, while it unveils interesting practical association patterns as the network dimensions go large. The load-balancing approach, combined with power control factors, significantly enhances system performance compared to conventional schemes and configurations with fixed factors.