We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of microfluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is induced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation step length along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.