In this work we present two contributions to cut mesh methods: a provably energy stable discontinuous Galerkin (DG) method with state redistribution and an entropy stable DG scheme for hyperbolic conservation laws. Cut meshes schemes use a simple unfitted background mesh of standard elements that is then cut by embedded boundaries to yield a hybrid mesh of cut and uncut elements. While these methods are a convenient means of mesh generation for complex geometries, the arbitrary size and shape of cut elements pose two significant challenges: the small cell problem--where the arbitrary size/shape of cut elements can severely restrict the CFL condition--and the construction of stable and accurate schemes on cut elements. We address the small cell problem using state redistribution and prove it can be combined with an energy stable scheme without damaging energy stability. For entropy stability, we construct hybridized summation-by-parts operators to reduce guaranteeing entropy stability to constructing exact, positive weight quadrature rules on cut elements. We construct such quadrature on cut elements using subtriangulations and Carathéodory pruning. We numerically verify the high-order accuracy and energy/entropy conservation and stability of our schemes using the acoustic wave, shallow water, and compressible Euler equations.