In this paper, we address modeling and control of a Hawkmoth Flapping Wing Micro Air Vehicle (MAV), and present a framework to systematically address critical control relevant tradeoffs associated with it. Nonlinear model of longitudinal dynamics of the MAV is considered, both in hover as well as in forward flight conditions. Averaging theory is used to convert the nonlinear time-varying, but periodic model into a nonlinear time-invariant one. The averaged model is then linearized at different flight conditions. The linearized models are then studied to address critical relevant questions. A novel H-infinity control methodology based on convex optimization-Generalized Mixed Sensitivity (GMS)-is presented that can handle wide range of control specifications (e.g., frequency-and time-domain closed loop properties at plant output and plant input). We show how critical closed loop multivariable properties can be shaped directly, based on specifications, using GMS Hierarchical control framework. Popular classically motivated controllers are designed and compared to illustrate the utility of GMS.