The talk will survey several design choices for finite-volume (FV) methods that will become candidates for a prototype 3D dynamical core on an adaptive cubed-sphere grid. The presentation will cover two parts. First, high-order finite volume methods for the shallow water equations on a gnomonic cubed-sphere grid are discussed. In particular, we focus on third- and fourth-order FV methods that can be paired with a variety of Riemann solvers for the numerical flux. The performance of these approaches is evaluated via the standard shallow water test suite. The second part of the talk addresses wave reflection issues that might arise on non-uniform non-conformal grids such as a statically nested grid or an adaptive mesh refinement (AMR) grid. We use a linear 1D wave analysis technique to shed light on the wave reflection properties of selected finite-volume methods. An outlook of our future AMR research will be given.