This paper investigates an in-band full-duplex (IBFD) unmanned aerial vehicle (UAV) communication system operating under realistic hardware impairments (HI). Although IBFD-UAV architectures offer strong potential for enhancing spectral efficiency and communication coverage, their practical deployment is fundamentally challenged by unavoidable hardware-induced distortions and residual self-interference (RSI). To counteract the negative effects of HI and RSI on system reliability, we derive exact and asymptotic expressions for the outage probability (OP) and throughput of the IBFD-UAV system (denoted as the IBFD-HI system). These expressions are general and can be adapted to ideal-hardware IBFD (IBFD-ID) and half-duplex (HD) systems with or without HI by appropriately modifying system parameters. To mitigate the detrimental effects of HI, RSI, and other performance-limiting factors, an optimal transmission power allocation scheme is recommended. Numerical results demonstrate that both HI and RSI dramatically degrade OP and throughput, while the proposed power optimization strategy effectively alleviates these impairments. Notably, an IBFD-HI system with optimal power allocation can even surpass an IBFD-ID system without power optimization, underscoring the practical importance of impairment-aware resource allocation. The analysis further examines the influence of key parameters, including HI level, RSI strength, data rate, UAV speed, operating frequency, and UAV positioning, on system behavior, from which several design insights are provided. All analytical findings are validated through Monte-Carlo simulations.