Cut-cell methods simplify mesh generation by using a Cartesian grid for the fluid domain and simply cutting out the solid body from the grid. However, this creates non-uniform cells near the solid boundary that often leads to cut-cell instabilities and small-cell issues which are particularly severe for (high-fidelity) flow calculations that avoid the use of artificial/numerical dissipation. As a result, existing non-dissipative cut-cell methods are mostly restricted to second-order accuracy, and ad hoc approaches (e.g. cell merging, flux redistribution, etc.) are commonly employed to address the small-cell problem. This study applies energy stability theory to derive cut-cell boundary treatments for hyperbolic, parabolic, and elliptic problems relevant to incompressible and compressible fluid flows. The resulting schemes are globally fourth-order accurate and ensure time stability without introducing numerical dissipation. The small-cell issue is addressed by including a constraint in the derivation that prevents the flux point (or cell) spacings from vanishing even when the grid points at the cut-cell boundary coincide. The derived cut-cell method addresses the small-cell issue by design and avoids the use of ad hoc approaches that compromise accuracy and are difficult to automate. The proposed cut-cell treatment uses finite-difference (FD) discretization, %with colocated variables for incompressible/compressible flows. An FD method is which are dimensionally-split by design, allowing the overall scheme for a two-/three-dimensional problem to be written as a combination of one-dimensional schemes. The theoretical stability proof developed for a one-dimensional problem is thus systematically extended to higher dimensions, providing a provably stable method. A dimensionally-split framework avoids geometry/solution reconstructions, improving the computational efficiency and making it easy to incorporate the method in an existing solver. The proposed method is applied to perform a series of inviscid and viscous flow simulations, including three-dimensional flow separation calculation over a bluff body where the grid around the cut-cell boundary is selectively refined using overset grids.