We develop a comprehensive framework to characterize and optimize the performance of a unmanned aerial vehicle (UAV)-assisted D2D network, where D2D transmissions underlay cellular transmissions. Different from conventional non-line-of-sight (NLoS) terrestrial transmissions, aerial transmissions are highly likely to experience line-of-sight (LoS). As such, characterizing the performance of mixed aerial-terrestrial networks with accurate fading models is critical to precise network performance characterization and resource optimization. We first characterize closed-form expressions for a variety of performance metrics such as frame decoding error probability (referred to as reliability), outage probability, and ergodic capacity of users. The terrestrial and aerial transmissions may experience either LoS Rician fading or NLoS Nakagami-m fading with a certain probability. Based on the derived expressions, we formulate a hierarchical bi-objective mixed-integer-nonlinear-programming (MINLP) problem to minimize the total transmit power of all users and maximize the aggregate throughput of D2D users subject to quality-of-service (QoS) measures (i.e., reliability and ergodic capacity) of cellular users. We model the proposed problem as a bi-partite one-to-many matching game. To solve this problem, we first obtain the optimal closed-form power allocations for each D2D and cellular user on any possible subchannel, and then incorporate them to devise efficient subchannel and power allocation algorithms. Complexity analysis of the proposed algorithms is presented. Numerical results verify the accuracy of our derived expressions and reveal the significance of aerial relays compared to ground relays in increasing the throughput of D2D pairs especially for distant D2D pairs.