In this work, we investigate rate-splitting multiple access (RSMA) for a multiuser downlink wireless network consisting of an unmanned aerial vehicle (UAV)-assisted base station (BS) that serves multiple ground users (GUs) simultaneously. Considering two different transmission regimes, namely infinite blocklength (IBL), and finite blocklength (FBL), we analyze the performance of the considered network under the effect of imperfections in channel state information (CSI) estimation and successive interference cancellation (SIC) with the probabilistic line of sight fading channels. For IBL transmission, we derive the closed-form expressions of the outage probability, throughput, and achievable ergodic rate at each GU. Furthermore, for short packet communication, the closed-form expressions of block error rate (BLER), goodput, and achievable ergodic rate are determined with FBL transmission. Moreover, for the FBL regime we also formulate an optimization problem that jointly optimizes the 3D-position of the UAV, power allocated to each user, and common rate distribution at each user to maximize the ergodic sum rate subject to the practical constraints such as maximum tolerable BLER, minimum private and total rate at each user. We then propose an alternating optimization-based iterative algorithm to solve the problem. Monte-Carlo simulations are used to verify the accuracy of derived analytical results and demonstrate the trade-off between transmit power and achievable BLER. In addition to this, the effectiveness of RSMA in UAV-assisted communication under IBL and FBL transmission regimes is also observed compared to non-orthogonal multiple access.